Central limit theorem

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id="toc-Independent_sequences-sublist" class="vector-toc-list"> <li id="toc-Classical_CLT" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classical_CLT"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Classical CLT</span> </div> </a> <ul id="toc-Classical_CLT-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lyapunov_CLT" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lyapunov_CLT"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Lyapunov CLT</span> </div> </a> <ul id="toc-Lyapunov_CLT-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lindeberg_(-Feller)_CLT" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lindeberg_(-Feller)_CLT"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Lindeberg (-Feller) CLT</span> </div> </a> <ul id="toc-Lindeberg_(-Feller)_CLT-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multidimensional_CLT" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multidimensional_CLT"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Multidimensional CLT</span> </div> </a> <ul id="toc-Multidimensional_CLT-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-The_Generalized_Central_Limit_Theorem" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#The_Generalized_Central_Limit_Theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>The Generalized Central Limit Theorem</span> </div> </a> <ul id="toc-The_Generalized_Central_Limit_Theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dependent_processes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Dependent_processes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Dependent processes</span> </div> </a> <button aria-controls="toc-Dependent_processes-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Dependent processes subsection</span> </button> <ul id="toc-Dependent_processes-sublist" class="vector-toc-list"> <li id="toc-CLT_under_weak_dependence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#CLT_under_weak_dependence"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>CLT under weak dependence</span> </div> </a> <ul id="toc-CLT_under_weak_dependence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Martingale_difference_CLT" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Martingale_difference_CLT"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Martingale difference CLT</span> </div> </a> <ul id="toc-Martingale_difference_CLT-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Remarks" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Remarks"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Remarks</span> </div> </a> <button aria-controls="toc-Remarks-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Remarks subsection</span> </button> <ul id="toc-Remarks-sublist" class="vector-toc-list"> <li id="toc-Proof_of_classical_CLT" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proof_of_classical_CLT"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Proof of classical CLT</span> </div> </a> <ul id="toc-Proof_of_classical_CLT-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Convergence_to_the_limit" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Convergence_to_the_limit"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Convergence to the limit</span> </div> </a> <ul id="toc-Convergence_to_the_limit-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Common_misconceptions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Common_misconceptions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Common misconceptions</span> </div> </a> <ul id="toc-Common_misconceptions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_the_law_of_large_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_the_law_of_large_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Relation to the law of large numbers</span> </div> </a> <ul id="toc-Relation_to_the_law_of_large_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Alternative_statements_of_the_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Alternative_statements_of_the_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Alternative statements of the theorem</span> </div> </a> <ul id="toc-Alternative_statements_of_the_theorem-sublist" class="vector-toc-list"> <li id="toc-Density_functions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Density_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5.1</span> <span>Density functions</span> </div> </a> <ul id="toc-Density_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Characteristic_functions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Characteristic_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5.2</span> <span>Characteristic functions</span> </div> </a> <ul id="toc-Characteristic_functions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Calculating_the_variance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Calculating_the_variance"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Calculating the variance</span> </div> </a> <ul id="toc-Calculating_the_variance-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Extensions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Extensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Extensions</span> </div> </a> <button aria-controls="toc-Extensions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Extensions subsection</span> </button> <ul id="toc-Extensions-sublist" class="vector-toc-list"> <li id="toc-Products_of_positive_random_variables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Products_of_positive_random_variables"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Products of positive random variables</span> </div> </a> <ul id="toc-Products_of_positive_random_variables-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Beyond_the_classical_framework" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Beyond_the_classical_framework"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Beyond the classical framework</span> </div> </a> <button aria-controls="toc-Beyond_the_classical_framework-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Beyond the classical framework subsection</span> </button> <ul id="toc-Beyond_the_classical_framework-sublist" class="vector-toc-list"> <li id="toc-Convex_body" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Convex_body"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Convex body</span> </div> </a> <ul id="toc-Convex_body-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lacunary_trigonometric_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lacunary_trigonometric_series"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Lacunary trigonometric series</span> </div> </a> <ul id="toc-Lacunary_trigonometric_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gaussian_polytopes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gaussian_polytopes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Gaussian polytopes</span> </div> </a> <ul id="toc-Gaussian_polytopes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linear_functions_of_orthogonal_matrices" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linear_functions_of_orthogonal_matrices"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Linear functions of orthogonal matrices</span> </div> </a> <ul id="toc-Linear_functions_of_orthogonal_matrices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subsequences" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subsequences"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Subsequences</span> </div> </a> <ul id="toc-Subsequences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Random_walk_on_a_crystal_lattice" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Random_walk_on_a_crystal_lattice"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.6</span> <span>Random walk on a crystal lattice</span> </div> </a> <ul id="toc-Random_walk_on_a_crystal_lattice-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications_and_examples" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications_and_examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Applications and examples</span> </div> </a> <button aria-controls="toc-Applications_and_examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Applications and examples subsection</span> </button> <ul id="toc-Applications_and_examples-sublist" class="vector-toc-list"> <li id="toc-Regression" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Regression"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Regression</span> </div> </a> <ul id="toc-Regression-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_illustrations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_illustrations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Other illustrations</span> </div> </a> <ul id="toc-Other_illustrations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Central limit theorem</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 41 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-41" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">41 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Sentrale_limietstelling" title="Sentrale limietstelling – Afrikaans" lang="af" hreflang="af" data-title="Sentrale limietstelling" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%A8%D8%B1%D9%87%D9%86%D8%A9_%D8%A7%D9%84%D9%86%D9%87%D8%A7%D9%8A%D8%A9_%D8%A7%D9%84%D9%85%D8%B1%D9%83%D8%B2%D9%8A%D8%A9" title="مبرهنة النهاية المركزية – Arabic" lang="ar" hreflang="ar" data-title="مبرهنة النهاية المركزية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teorema_de_la_llende_central" title="Teorema de la llende central – Asturian" lang="ast" hreflang="ast" data-title="Teorema de la llende central" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A6%D1%8D%D0%BD%D1%82%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%BB%D1%96%D0%BC%D1%96%D1%82%D0%B0%D0%B2%D0%B0%D1%8F_%D1%82%D1%8D%D0%B0%D1%80%D1%8D%D0%BC%D0%B0" title="Цэнтральная лімітавая тэарэма – Belarusian" lang="be" hreflang="be" data-title="Цэнтральная лімітавая тэарэма" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teorema_del_l%C3%ADmit_central" title="Teorema del límit central – Catalan" lang="ca" hreflang="ca" data-title="Teorema del límit central" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Centr%C3%A1ln%C3%AD_limitn%C3%AD_v%C4%9Bta" title="Centrální limitní věta – Czech" lang="cs" hreflang="cs" data-title="Centrální limitní věta" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Zentraler_Grenzwertsatz" title="Zentraler Grenzwertsatz – German" lang="de" hreflang="de" data-title="Zentraler Grenzwertsatz" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Tsentraalne_piirteoreem" title="Tsentraalne piirteoreem – Estonian" lang="et" hreflang="et" data-title="Tsentraalne piirteoreem" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%B9%CE%BA%CE%BF%CF%8D_%CE%BF%CF%81%CE%AF%CE%BF%CF%85" title="Θεώρημα κεντρικού ορίου – Greek" lang="el" hreflang="el" data-title="Θεώρημα κεντρικού ορίου" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teorema_del_l%C3%ADmite_central" title="Teorema del límite central – Spanish" lang="es" hreflang="es" data-title="Teorema del límite central" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Limitearen_teorema_zentral" title="Limitearen teorema zentral – Basque" lang="eu" hreflang="eu" data-title="Limitearen teorema zentral" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%B6%DB%8C%D9%87_%D8%AD%D8%AF_%D9%85%D8%B1%DA%A9%D8%B2%DB%8C" title="قضیه حد مرکزی – Persian" lang="fa" hreflang="fa" data-title="قضیه حد مرکزی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_central_limite" title="Théorème central limite – French" lang="fr" hreflang="fr" data-title="Théorème central limite" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teorema_central_do_l%C3%ADmite" title="Teorema central do límite – Galician" lang="gl" hreflang="gl" data-title="Teorema central do límite" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A4%91%EC%8B%AC_%EA%B7%B9%ED%95%9C_%EC%A0%95%EB%A6%AC" title="중심 극한 정리 – Korean" lang="ko" hreflang="ko" data-title="중심 극한 정리" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teorema_limit_pusat" title="Teorema limit pusat – Indonesian" lang="id" hreflang="id" data-title="Teorema limit pusat" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is 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data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Centr%C4%81l%C4%81_robe%C5%BEteor%C4%93ma" title="Centrālā robežteorēma – Latvian" lang="lv" hreflang="lv" data-title="Centrālā robežteorēma" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Centr%C3%A1lis_hat%C3%A1reloszl%C3%A1s-t%C3%A9tel" title="Centrális határeloszlás-tétel – Hungarian" lang="hu" hreflang="hu" data-title="Centrális határeloszlás-tétel" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BD%D1%82%D1%80%D0%B0%D0%BB%D0%BD%D0%B0_%D0%B3%D1%80%D0%B0%D0%BD%D0%B8%D1%87%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Централна гранична теорема – Macedonian" lang="mk" hreflang="mk" data-title="Централна гранична теорема" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Centrale_limietstelling" title="Centrale limietstelling – Dutch" lang="nl" hreflang="nl" data-title="Centrale limietstelling" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%B8%AD%E5%BF%83%E6%A5%B5%E9%99%90%E5%AE%9A%E7%90%86" title="中心極限定理 – Japanese" lang="ja" hreflang="ja" data-title="中心極限定理" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Sentralgrenseteoremet" title="Sentralgrenseteoremet – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Sentralgrenseteoremet" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Centralne_twierdzenie_graniczne" title="Centralne twierdzenie graniczne – Polish" lang="pl" hreflang="pl" data-title="Centralne twierdzenie graniczne" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teorema_central_do_limite" title="Teorema central do limite – Portuguese" lang="pt" hreflang="pt" data-title="Teorema central do limite" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BD%D1%82%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Центральная предельная теорема – Russian" lang="ru" hreflang="ru" data-title="Центральная предельная теорема" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Teorema_Q%C3%ABndrore_Limite" title="Teorema Qëndrore Limite – Albanian" lang="sq" hreflang="sq" data-title="Teorema Qëndrore Limite" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Central_limit_theorem" title="Central limit theorem – Simple English" lang="en-simple" hreflang="en-simple" data-title="Central limit theorem" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BD%D1%82%D1%80%D0%B0%D0%BB%D0%BD%D0%B0_%D0%B3%D1%80%D0%B0%D0%BD%D0%B8%D1%87%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Централна гранична теорема – Serbian" lang="sr" hreflang="sr" data-title="Централна гранична теорема" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Central_limit_theorem" title="Central limit theorem – Sundanese" lang="su" hreflang="su" data-title="Central limit theorem" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Keskeinen_raja-arvolause" title="Keskeinen raja-arvolause – Finnish" lang="fi" hreflang="fi" data-title="Keskeinen raja-arvolause" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Centrala_gr%C3%A4nsv%C3%A4rdessatsen" title="Centrala gränsvärdessatsen – Swedish" lang="sv" hreflang="sv" data-title="Centrala gränsvärdessatsen" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Merkez%C3%AE_limit_teoremi" title="Merkezî limit teoremi – Turkish" lang="tr" hreflang="tr" data-title="Merkezî limit teoremi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BD%D1%82%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0_%D0%B3%D1%80%D0%B0%D0%BD%D0%B8%D1%87%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Центральна гранична теорема – Ukrainian" lang="uk" hreflang="uk" data-title="Центральна гранична теорема" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%B1%DA%A9%D8%B2%DB%8C_%D8%AD%D8%AF_%D9%85%D8%B3%D8%A6%D9%84%DB%81_%D8%A7%D8%AB%D8%A8%D8%A7%D8%AA%DB%8C" title="مرکزی حد مسئلہ اثباتی – Urdu" lang="ur" hreflang="ur" data-title="مرکزی حد مسئلہ اثباتی" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_gi%E1%BB%9Bi_h%E1%BA%A1n_trung_t%C3%A2m" title="Định lý giới hạn trung tâm – Vietnamese" lang="vi" hreflang="vi" data-title="Định lý giới hạn trung tâm" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E4%B8%AD%E5%BF%83%E6%9E%81%E9%99%90%E5%AE%9A%E7%90%86" title="中心极限定理 – Wu" lang="wuu" hreflang="wuu" data-title="中心极限定理" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%B8%AD%E5%A4%AE%E6%A5%B5%E9%99%90%E5%AE%9A%E7%90%86" title="中央極限定理 – Cantonese" lang="yue" hreflang="yue" data-title="中央極限定理" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%B8%AD%E5%BF%83%E6%9E%81%E9%99%90%E5%AE%9A%E7%90%86" title="中心极限定理 – Chinese" lang="zh" hreflang="zh" data-title="中心极限定理" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div 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<div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Fundamental theorem in probability theory and statistics</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox vcard"><caption class="infobox-title fn" style="padding-bottom:0.2em;">Central Limit Theorem</caption><tbody><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:IllustrationCentralTheorem.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/IllustrationCentralTheorem.png/220px-IllustrationCentralTheorem.png" decoding="async" width="220" height="93" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/IllustrationCentralTheorem.png/330px-IllustrationCentralTheorem.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7b/IllustrationCentralTheorem.png/440px-IllustrationCentralTheorem.png 2x" data-file-width="2771" data-file-height="1166" /></a></span></td></tr><tr><th scope="row" class="infobox-label">Type</th><td class="infobox-data"><a href="/wiki/Theorem" title="Theorem">Theorem</a></td></tr><tr><th scope="row" class="infobox-label">Field</th><td class="infobox-data"><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></td></tr><tr><th scope="row" class="infobox-label">Statement</th><td class="infobox-data">The scaled sum of a sequence of <a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">i.i.d. random variables</a> with finite positive <a href="/wiki/Variance" title="Variance">variance</a> converges in distribution to the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>.</td></tr><tr><th scope="row" class="infobox-label">Generalizations</th><td class="infobox-data"><a href="/wiki/Lindeberg%27s_condition" title="Lindeberg's condition"> Lindeberg's CLT</a></td></tr></tbody></table> <p>In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, the <b>central limit theorem</b> (<b>CLT</b>) states that, under appropriate conditions, the <a href="/wiki/Probability_distribution" title="Probability distribution">distribution</a> of a normalized version of the sample mean converges to a <a href="/wiki/Normal_distribution#Standard_normal_distribution" title="Normal distribution">standard normal distribution</a>. This holds even if the original variables themselves are not <a href="/wiki/Normal_distribution" title="Normal distribution">normally distributed</a>. There are several versions of the CLT, each applying in the context of different conditions. </p><p>The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions. </p><p>This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern form it was only precisely stated as late as 1920.<sup id="cite_ref-FOOTNOTEFischer2011[[Category:Wikipedia_articles_needing_page_number_citations_from_July_2023]]<sup_class="noprint_Inline-Template_"_style="white-space:nowrap;">&#91;<i>[[Wikipedia:Citing_sources|<span_title="This_citation_requires_a_reference_to_the_specific_page_or_range_of_pages_in_which_the_material_appears.&#32;(July_2023)">page&nbsp;needed</span>]]</i>&#93;</sup>_1-0" class="reference"><a href="#cite_note-FOOTNOTEFischer2011[[Category:Wikipedia_articles_needing_page_number_citations_from_July_2023]]<sup_class="noprint_Inline-Template_"_style="white-space:nowrap;">&#91;<i>[[Wikipedia:Citing_sources|<span_title="This_citation_requires_a_reference_to_the_specific_page_or_range_of_pages_in_which_the_material_appears.&#32;(July_2023)">page&nbsp;needed</span>]]</i>&#93;</sup>-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, the CLT can be stated as: let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},\dots ,X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},\dots ,X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/197911bbfe9fe9540571f88763a0064df10686ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.312ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},\dots ,X_{n}}"></span> denote a <a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">statistical sample</a> of size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> from a population with <a href="/wiki/Expected_value" title="Expected value">expected value</a> (average) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> and finite positive <a href="/wiki/Variance" title="Variance">variance</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}"></span>, and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ab9d44cdd15adc4e2e00c9b282832eefe22d2dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.198ex; height:2.843ex;" alt="{\displaystyle {\bar {X}}_{n}}"></span> denote the sample mean (which is itself a <a href="/wiki/Random_variable" title="Random variable">random variable</a>). Then the <a href="/wiki/Convergence_of_random_variables#Convergence_in_distribution" title="Convergence of random variables">limit as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\to \infty }"></span> of the distribution</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\bar {X}}_{n}-\mu ){\sqrt {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\bar {X}}_{n}-\mu ){\sqrt {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0140363b34a41babc747665cd5a1d09eaeb8bc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.58ex; height:3.176ex;" alt="{\displaystyle ({\bar {X}}_{n}-\mu ){\sqrt {n}}}"></span> is a normal distribution with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> and variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}}"></span>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>In other words, suppose that a large sample of <a href="/wiki/Random_variate" title="Random variate">observations</a> is obtained, each observation being randomly produced in a way that does not depend on the values of the other observations, and the average (<a href="/wiki/Arithmetic_mean" title="Arithmetic mean">arithmetic mean</a>) of the observed values is computed. If this procedure is performed many times, resulting in a collection of observed averages, the central limit theorem says that if the sample size is large enough, the <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> of these averages will closely approximate a normal distribution. </p><p>The central limit theorem has several variants. In its common form, the random variables must be <a href="/wiki/Independent_and_identically_distributed" class="mw-redirect" title="Independent and identically distributed">independent and identically distributed</a> (i.i.d.). This requirement can be weakened; convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations if they comply with certain conditions. </p><p>The earliest version of this theorem, that the normal distribution may be used as an approximation to the <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a>, is the <a href="/wiki/De_Moivre%E2%80%93Laplace_theorem" title="De Moivre–Laplace theorem">de Moivre–Laplace theorem</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Independent_sequences">Independent sequences</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=1" title="Edit section: Independent sequences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:IllustrationCentralTheorem.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/IllustrationCentralTheorem.png/310px-IllustrationCentralTheorem.png" decoding="async" width="310" height="130" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/IllustrationCentralTheorem.png/465px-IllustrationCentralTheorem.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7b/IllustrationCentralTheorem.png/620px-IllustrationCentralTheorem.png 2x" data-file-width="2771" data-file-height="1166" /></a><figcaption>Whatever the form of the population distribution, the sampling distribution tends to a Gaussian, and its dispersion is given by the central limit theorem.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Classical_CLT">Classical CLT</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=2" title="Edit section: Classical CLT"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X_{1},\ldots ,X_{n}}\}"> <semantics> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <mo fence="false" stretchy="false">}</mo> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X_{1},\ldots ,X_{n}}\}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f42c7722167682ff6397a584faba39d0e77276e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.624ex; height:2.843ex;" alt="{\displaystyle \{X_{1},\ldots ,X_{n}}\}"></span> be a sequence of <a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">i.i.d. random variables</a> having a distribution with <a href="/wiki/Expected_value" title="Expected value">expected value</a> given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> and finite <a href="/wiki/Variance" title="Variance">variance</a> given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c82112327f2b74f6bf7ba72fe64621a7ae9064de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.032ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}.}"></span> Suppose we are interested in the <a href="/wiki/Sample_mean" class="mw-redirect" title="Sample mean">sample average</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}_{n}\equiv {\frac {X_{1}+\cdots +X_{n}}{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}_{n}\equiv {\frac {X_{1}+\cdots +X_{n}}{n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da94013ecb1dbc10d9a54c8eb669a3a87171582e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.305ex; height:5.176ex;" alt="{\displaystyle {\bar {X}}_{n}\equiv {\frac {X_{1}+\cdots +X_{n}}{n}}.}"></span> </p><p>By the <a href="/wiki/Law_of_large_numbers" title="Law of large numbers">law of large numbers</a>, the sample average <a href="/wiki/Almost_sure_convergence" class="mw-redirect" title="Almost sure convergence">converges almost surely</a> (and therefore also <a href="/wiki/Convergence_in_probability" class="mw-redirect" title="Convergence in probability">converges in probability</a>) to the expected value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\to \infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8a34a9f62668de90200a6cbde865c27af2cdbb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.979ex; height:1.843ex;" alt="{\displaystyle n\to \infty .}"></span> </p><p>The classical central limit theorem describes the size and the distributional form of the stochastic fluctuations around the deterministic number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> during this convergence. More precisely, it states that as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> gets larger, the distribution of the normalized mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1f533faaeb8931b80712ddb6361d4814c2be02a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.58ex; height:3.176ex;" alt="{\displaystyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}"></span>, i.e. the difference between the sample average <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ab9d44cdd15adc4e2e00c9b282832eefe22d2dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.198ex; height:2.843ex;" alt="{\displaystyle {\bar {X}}_{n}}"></span> and its limit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e7e1ef161a49a22b500d63307460ad92eeb6a16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.049ex; height:2.176ex;" alt="{\displaystyle \mu ,}"></span> scaled by the factor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a2994734eae382ce30100fb17b9447fd8e99f81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.331ex; height:3.009ex;" alt="{\displaystyle {\sqrt {n}}}"></span>, approaches the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> and variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c82112327f2b74f6bf7ba72fe64621a7ae9064de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.032ex; height:2.676ex;" alt="{\displaystyle \sigma ^{2}.}"></span> For large enough <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397bfafc701afdf14c2743278a097f6f2957eabb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.042ex; height:2.009ex;" alt="{\displaystyle n,}"></span> the distribution of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ab9d44cdd15adc4e2e00c9b282832eefe22d2dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.198ex; height:2.843ex;" alt="{\displaystyle {\bar {X}}_{n}}"></span> gets arbitrarily close to the normal distribution with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> and variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}/n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}/n.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d48d9aeca39ca2771b0b7e941dd17f652df8f3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.589ex; height:3.176ex;" alt="{\displaystyle \sigma ^{2}/n.}"></span> </p><p>The usefulness of the theorem is that the distribution of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1f533faaeb8931b80712ddb6361d4814c2be02a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.58ex; height:3.176ex;" alt="{\displaystyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}"></span> approaches normality regardless of the shape of the distribution of the individual <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d8cf06c3e80129deea779a3d738464b1990906c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.509ex;" alt="{\displaystyle X_{i}.}"></span> Formally, the theorem can be stated as follows: </p> <style data-mw-deduplicate="TemplateStyles:r1110004140">.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}</style><div class="math_theorem" style=""> <p><strong class="theorem-name">Lindeberg–Lévy CLT</strong><span class="theoreme-tiret"> — </span>Suppose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},X_{3}\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},X_{3}\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd6598526ef19d380b128f3aacff83367b5f35c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.114ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},X_{3}\ldots }"></span> is a sequence of <a href="/wiki/Independent_and_identically_distributed" class="mw-redirect" title="Independent and identically distributed">i.i.d.</a> random variables with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} [X_{i}]=\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} [X_{i}]=\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4cb0f7a2cebcaf8b28056deff750e7a0e1d34f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.101ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} [X_{i}]=\mu }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Var} [X_{i}]=\sigma ^{2}<\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} [X_{i}]=\sigma ^{2}<\infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76c2379b589ea26ef8d17aaeab2d9899f30f1299" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.387ex; height:3.176ex;" alt="{\displaystyle \operatorname {Var} [X_{i}]=\sigma ^{2}<\infty .}"></span> Then, as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> approaches infinity, the random variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1f533faaeb8931b80712ddb6361d4814c2be02a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.58ex; height:3.176ex;" alt="{\displaystyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}"></span> <a href="/wiki/Convergence_in_distribution" class="mw-redirect" title="Convergence in distribution">converge in distribution</a> to a <a href="/wiki/Normal_distribution" title="Normal distribution">normal</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}(0,\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}(0,\sigma ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a12e4999caaf1154cee3440edde18c9e5f66a8da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:8.727ex; height:3.176ex;" alt="{\displaystyle {\mathcal {N}}(0,\sigma ^{2})}"></span>:<sup id="cite_ref-FOOTNOTEBillingsley1995357_4-0" class="reference"><a href="#cite_note-FOOTNOTEBillingsley1995357-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}\left({\bar {X}}_{n}-\mu \right)\mathrel {\overset {d}{\longrightarrow }} {\mathcal {N}}\left(0,\sigma ^{2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-REL"> <mover> <mo stretchy="false">⟶</mo> <mi>d</mi> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}}\left({\bar {X}}_{n}-\mu \right)\mathrel {\overset {d}{\longrightarrow }} {\mathcal {N}}\left(0,\sigma ^{2}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3752ee1b5177b8d87fb1cde66b4b4eef328bb142" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.791ex; height:4.343ex;" alt="{\displaystyle {\sqrt {n}}\left({\bar {X}}_{n}-\mu \right)\mathrel {\overset {d}{\longrightarrow }} {\mathcal {N}}\left(0,\sigma ^{2}\right).}"></span> </p> </div> <p>In the case <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma >0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma >0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/135c308f5cd7c6374f738b618f1c7e09afe129f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.237ex; height:2.509ex;" alt="{\displaystyle \sigma >0,}"></span> convergence in distribution means that the <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution functions</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1f533faaeb8931b80712ddb6361d4814c2be02a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.58ex; height:3.176ex;" alt="{\displaystyle {\sqrt {n}}({\bar {X}}_{n}-\mu )}"></span> converge pointwise to the cdf of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}(0,\sigma ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}(0,\sigma ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a12e4999caaf1154cee3440edde18c9e5f66a8da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:8.727ex; height:3.176ex;" alt="{\displaystyle {\mathcal {N}}(0,\sigma ^{2})}"></span> distribution: for every real number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47989a9b66a4ea8a0ec19e8159749fce8a9a8ca8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.735ex; height:2.009ex;" alt="{\displaystyle z,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }\mathbb {P} \left[{\sqrt {n}}({\bar {X}}_{n}-\mu )\leq z\right]=\lim _{n\to \infty }\mathbb {P} \left[{\frac {{\sqrt {n}}({\bar {X}}_{n}-\mu )}{\sigma }}\leq {\frac {z}{\sigma }}\right]=\Phi \left({\frac {z}{\sigma }}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>z</mi> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> <mi>σ</mi> </mfrac> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mi>σ</mi> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mi mathvariant="normal">Φ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }\mathbb {P} \left[{\sqrt {n}}({\bar {X}}_{n}-\mu )\leq z\right]=\lim _{n\to \infty }\mathbb {P} \left[{\frac {{\sqrt {n}}({\bar {X}}_{n}-\mu )}{\sigma }}\leq {\frac {z}{\sigma }}\right]=\Phi \left({\frac {z}{\sigma }}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/defd4cf70972fa6a76a8570fee6551f4cb7d70b8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:67.269ex; height:7.509ex;" alt="{\displaystyle \lim _{n\to \infty }\mathbb {P} \left[{\sqrt {n}}({\bar {X}}_{n}-\mu )\leq z\right]=\lim _{n\to \infty }\mathbb {P} \left[{\frac {{\sqrt {n}}({\bar {X}}_{n}-\mu )}{\sigma }}\leq {\frac {z}{\sigma }}\right]=\Phi \left({\frac {z}{\sigma }}\right),}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fbffdd4f2cf1c6e8daaddfddd00677b16f12809" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.576ex; height:2.843ex;" alt="{\displaystyle \Phi (z)}"></span> is the standard normal cdf evaluated at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd7f273b229260c8fe9aa42378b0471336394cc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.735ex; height:1.676ex;" alt="{\displaystyle z.}"></span> The convergence is uniform in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> in the sense that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }\;\sup _{z\in \mathbb {R} }\;\left|\mathbb {P} \left[{\sqrt {n}}({\bar {X}}_{n}-\mu )\leq z\right]-\Phi \left({\frac {z}{\sigma }}\right)\right|=0~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mspace width="thickmathspace" /> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </munder> <mspace width="thickmathspace" /> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>z</mi> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <mi mathvariant="normal">Φ</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mi>σ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }\;\sup _{z\in \mathbb {R} }\;\left|\mathbb {P} \left[{\sqrt {n}}({\bar {X}}_{n}-\mu )\leq z\right]-\Phi \left({\frac {z}{\sigma }}\right)\right|=0~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/835addcb3ec37594d1e9a6a78c0373a5e7b2eddc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:46.981ex; height:5.509ex;" alt="{\displaystyle \lim _{n\to \infty }\;\sup _{z\in \mathbb {R} }\;\left|\mathbb {P} \left[{\sqrt {n}}({\bar {X}}_{n}-\mu )\leq z\right]-\Phi \left({\frac {z}{\sigma }}\right)\right|=0~,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sup }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">sup</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/266a03117c05ef2558c304b1dbb5f319fcd56fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.501ex; height:2.009ex;" alt="{\displaystyle \sup }"></span> denotes the least upper bound (or <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">supremum</a>) of the set.<sup id="cite_ref-FOOTNOTEBauer2001199Theorem_30.13_5-0" class="reference"><a href="#cite_note-FOOTNOTEBauer2001199Theorem_30.13-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Lyapunov_CLT">Lyapunov CLT</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=3" title="Edit section: Lyapunov CLT"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In this variant of the central limit theorem the random variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70db743cc2ba4453965ff0bdd6d6ef4f6d0ec699" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\textstyle X_{i}}"></span> have to be independent, but not necessarily identically distributed. The theorem also requires that random variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left|X_{i}\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>|</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left|X_{i}\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4576839a2dde8200ec639c41d60f25dc5e311daa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.018ex; height:2.843ex;" alt="{\textstyle \left|X_{i}\right|}"></span> have <a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">moments</a> of some order <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (2+\delta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (2+\delta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2df82efa92af67a0e93dd48e5079860e949d715a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.861ex; height:2.843ex;" alt="{\textstyle (2+\delta )}"></span>,</span> and that the rate of growth of these moments is limited by the Lyapunov condition given below. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> <p><strong class="theorem-name">Lyapunov CLT<sup id="cite_ref-FOOTNOTEBillingsley1995362_6-0" class="reference"><a href="#cite_note-FOOTNOTEBillingsley1995362-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></strong><span class="theoreme-tiret"> — </span>Suppose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \{X_{1},\ldots ,X_{n},\ldots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \{X_{1},\ldots ,X_{n},\ldots \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8cf831d0f6051f596823e3ab16dec9b32c3605b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.382ex; height:2.843ex;" alt="{\textstyle \{X_{1},\ldots ,X_{n},\ldots \}}"></span> is a sequence of independent random variables, each with finite expected value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mu _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mu _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fd49d4df5ae6cae38aaec73f22e4eaacb29b253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.201ex; height:2.176ex;" alt="{\textstyle \mu _{i}}"></span> and variance <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sigma _{i}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sigma _{i}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83678ff991d38b63c52a71d9b4c3e129b9d98ff1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.385ex; height:3.176ex;" alt="{\textstyle \sigma _{i}^{2}}"></span>.</span> Define <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{n}^{2}=\sum _{i=1}^{n}\sigma _{i}^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{n}^{2}=\sum _{i=1}^{n}\sigma _{i}^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1df16c0efa2304bd4f4ca88a2bbfef75c13374b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.181ex; height:6.843ex;" alt="{\displaystyle s_{n}^{2}=\sum _{i=1}^{n}\sigma _{i}^{2}.}"></span> </p><p>If for some <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \delta >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>δ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \delta >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e503a231ad333c0bd9a2dc7d48d8848076423356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\textstyle \delta >0}"></span>,</span> <i>Lyapunov’s condition</i> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }\;{\frac {1}{s_{n}^{2+\delta }}}\,\sum _{i=1}^{n}\operatorname {E} \left[\left|X_{i}-\mu _{i}\right|^{2+\delta }\right]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mi>δ</mi> </mrow> </msubsup> </mfrac> </mrow> <mspace width="thinmathspace" /> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mrow> <mo>|</mo> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mi>δ</mi> </mrow> </msup> <mo>]</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }\;{\frac {1}{s_{n}^{2+\delta }}}\,\sum _{i=1}^{n}\operatorname {E} \left[\left|X_{i}-\mu _{i}\right|^{2+\delta }\right]=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f06b7a3309fd45005cce7a4e0b15ca3758f662f5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.994ex; height:6.843ex;" alt="{\displaystyle \lim _{n\to \infty }\;{\frac {1}{s_{n}^{2+\delta }}}\,\sum _{i=1}^{n}\operatorname {E} \left[\left|X_{i}-\mu _{i}\right|^{2+\delta }\right]=0}"></span> is satisfied, then a sum of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {X_{i}-\mu _{i}}{s_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {X_{i}-\mu _{i}}{s_{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e74b2c70018625d030bc0da067e9fbc0bae4921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.716ex; height:4.176ex;" alt="{\textstyle {\frac {X_{i}-\mu _{i}}{s_{n}}}}"></span> converges in distribution to a standard normal random variable, as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\textstyle n}"></span> goes to infinity: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{s_{n}}}\,\sum _{i=1}^{n}\left(X_{i}-\mu _{i}\right)\mathrel {\overset {d}{\longrightarrow }} {\mathcal {N}}(0,1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mspace width="thinmathspace" /> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-REL"> <mover> <mo stretchy="false">⟶</mo> <mi>d</mi> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{s_{n}}}\,\sum _{i=1}^{n}\left(X_{i}-\mu _{i}\right)\mathrel {\overset {d}{\longrightarrow }} {\mathcal {N}}(0,1).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14316bd39542fb71d72341e143c5a311ae21786b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.422ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{s_{n}}}\,\sum _{i=1}^{n}\left(X_{i}-\mu _{i}\right)\mathrel {\overset {d}{\longrightarrow }} {\mathcal {N}}(0,1).}"></span> </p> </div> <p>In practice it is usually easiest to check Lyapunov's condition for <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \delta =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>δ</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \delta =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0519d0fea3348d08426b0d99a7af0e13f938aea7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\textstyle \delta =1}"></span>.</span> </p><p>If a sequence of random variables satisfies Lyapunov's condition, then it also satisfies Lindeberg's condition. The converse implication, however, does not hold. </p> <div class="mw-heading mw-heading3"><h3 id="Lindeberg_(-Feller)_CLT"><span id="Lindeberg_.28-Feller.29_CLT"></span>Lindeberg (-Feller) CLT</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=4" title="Edit section: Lindeberg (-Feller) CLT"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Lindeberg%27s_condition" title="Lindeberg's condition">Lindeberg's condition</a></div> <p>In the same setting and with the same notation as above, the Lyapunov condition can be replaced with the following weaker one (from <a href="/wiki/Jarl_Waldemar_Lindeberg" title="Jarl Waldemar Lindeberg">Lindeberg</a> in 1920). </p><p>Suppose that for every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varepsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \varepsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e73828dc02cc2d6974733344381d188455ab3d55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\textstyle \varepsilon >0}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{\frac {1}{s_{n}^{2}}}\sum _{i=1}^{n}\operatorname {E} \left[(X_{i}-\mu _{i})^{2}\cdot \mathbf {1} _{\left\{\left|X_{i}-\mu _{i}\right|>\varepsilon s_{n}\right\}}\right]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mrow> <mrow> <mo>|</mo> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mo>></mo> <mi>ε</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>}</mo> </mrow> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {1}{s_{n}^{2}}}\sum _{i=1}^{n}\operatorname {E} \left[(X_{i}-\mu _{i})^{2}\cdot \mathbf {1} _{\left\{\left|X_{i}-\mu _{i}\right|>\varepsilon s_{n}\right\}}\right]=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfd10d152dd578ac2a2fa674a084bd7b03b95b1b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.814ex; height:6.843ex;" alt="{\displaystyle \lim _{n\to \infty }{\frac {1}{s_{n}^{2}}}\sum _{i=1}^{n}\operatorname {E} \left[(X_{i}-\mu _{i})^{2}\cdot \mathbf {1} _{\left\{\left|X_{i}-\mu _{i}\right|>\varepsilon s_{n}\right\}}\right]=0}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {1} _{\{\ldots \}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathbf {1} _{\{\ldots \}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d8bb3f9d0effd91db23de8e5c25323be916bdda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.138ex; height:3.009ex;" alt="{\textstyle \mathbf {1} _{\{\ldots \}}}"></span> is the <a href="/wiki/Indicator_function" title="Indicator function">indicator function</a>. Then the distribution of the standardized sums <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{s_{n}}}\sum _{i=1}^{n}\left(X_{i}-\mu _{i}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{s_{n}}}\sum _{i=1}^{n}\left(X_{i}-\mu _{i}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/455361b7063b96cc68099041e69a3461ffca46d2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.849ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{s_{n}}}\sum _{i=1}^{n}\left(X_{i}-\mu _{i}\right)}"></span> converges towards the standard normal distribution <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {N}}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathcal {N}}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba37ff02211fb81c813065b489b78ec9df951e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:7.505ex; height:3.009ex;" alt="{\textstyle {\mathcal {N}}(0,1)}"></span>.</span> </p> <div class="mw-heading mw-heading3"><h3 id="Multidimensional_CLT">Multidimensional CLT</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=5" title="Edit section: Multidimensional CLT"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Proofs that use characteristic functions can be extended to cases where each individual <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {X} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathbf {X} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1c5b312b59eec02027f69297c93d303697233eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.819ex; height:2.509ex;" alt="{\textstyle \mathbf {X} _{i}}"></span> is a <a href="/wiki/Random_vector" class="mw-redirect" title="Random vector">random vector</a> in <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbb {R} ^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathbb {R} ^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc5ce44a3fa7ec45276820499f4fdcbb89e12a7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.767ex; height:2.676ex;" alt="{\textstyle \mathbb {R} ^{k}}"></span>,</span> with mean vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\boldsymbol {\mu }}=\operatorname {E} [\mathbf {X} _{i}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\boldsymbol {\mu }}=\operatorname {E} [\mathbf {X} _{i}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2931b016ce578d17578ee3cdffeb31852446873" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.44ex; height:2.843ex;" alt="{\textstyle {\boldsymbol {\mu }}=\operatorname {E} [\mathbf {X} _{i}]}"></span> and <a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrix</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {\Sigma } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathbf {\Sigma } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ae7bb00bd54a3671be6b1f2b3a067462c91aa63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\textstyle \mathbf {\Sigma } }"></span> (among the components of the vector), and these random vectors are independent and identically distributed. The multidimensional central limit theorem states that when scaled, sums converge to a <a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">multivariate normal distribution</a>.<sup id="cite_ref-vanderVaart_7-0" class="reference"><a href="#cite_note-vanderVaart-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Summation of these vectors is done component-wise. </p><p>For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,2,3,\ldots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,2,3,\ldots ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75890aaf87574f07064dd7effe29c1ace708bead" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.247ex; height:2.509ex;" alt="{\displaystyle i=1,2,3,\ldots ,}"></span> let <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} _{i}={\begin{bmatrix}X_{i}^{(1)}\\\vdots \\X_{i}^{(k)}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {X} _{i}={\begin{bmatrix}X_{i}^{(1)}\\\vdots \\X_{i}^{(k)}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5849f8855c39fb6f72ec0bb0af6f027e9b764557" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.262ex; margin-bottom: -0.243ex; width:14.135ex; height:12.176ex;" alt="{\displaystyle \mathbf {X} _{i}={\begin{bmatrix}X_{i}^{(1)}\\\vdots \\X_{i}^{(k)}\end{bmatrix}}}"></span> be independent random vectors. The sum of the random vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} _{1},\ldots ,\mathbf {X} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {X} _{1},\ldots ,\mathbf {X} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ed48d149cb70706fe64e27eb403783b9163a686" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.49ex; height:2.509ex;" alt="{\displaystyle \mathbf {X} _{1},\ldots ,\mathbf {X} _{n}}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}\mathbf {X} _{i}={\begin{bmatrix}X_{1}^{(1)}\\\vdots \\X_{1}^{(k)}\end{bmatrix}}+{\begin{bmatrix}X_{2}^{(1)}\\\vdots \\X_{2}^{(k)}\end{bmatrix}}+\cdots +{\begin{bmatrix}X_{n}^{(1)}\\\vdots \\X_{n}^{(k)}\end{bmatrix}}={\begin{bmatrix}\sum _{i=1}^{n}X_{i}^{(1)}\\\vdots \\\sum _{i=1}^{n}X_{i}^{(k)}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}\mathbf {X} _{i}={\begin{bmatrix}X_{1}^{(1)}\\\vdots \\X_{1}^{(k)}\end{bmatrix}}+{\begin{bmatrix}X_{2}^{(1)}\\\vdots \\X_{2}^{(k)}\end{bmatrix}}+\cdots +{\begin{bmatrix}X_{n}^{(1)}\\\vdots \\X_{n}^{(k)}\end{bmatrix}}={\begin{bmatrix}\sum _{i=1}^{n}X_{i}^{(1)}\\\vdots \\\sum _{i=1}^{n}X_{i}^{(k)}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c25add12bc8b8482b3465848a882be2c756861a0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.262ex; margin-bottom: -0.243ex; width:62.612ex; height:12.176ex;" alt="{\displaystyle \sum _{i=1}^{n}\mathbf {X} _{i}={\begin{bmatrix}X_{1}^{(1)}\\\vdots \\X_{1}^{(k)}\end{bmatrix}}+{\begin{bmatrix}X_{2}^{(1)}\\\vdots \\X_{2}^{(k)}\end{bmatrix}}+\cdots +{\begin{bmatrix}X_{n}^{(1)}\\\vdots \\X_{n}^{(k)}\end{bmatrix}}={\begin{bmatrix}\sum _{i=1}^{n}X_{i}^{(1)}\\\vdots \\\sum _{i=1}^{n}X_{i}^{(k)}\end{bmatrix}}}"></span> and their average is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {{\bar {X}}_{n}} ={\begin{bmatrix}{\bar {X}}_{i}^{(1)}\\\vdots \\{\bar {X}}_{i}^{(k)}\end{bmatrix}}={\frac {1}{n}}\sum _{i=1}^{n}\mathbf {X} _{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">X</mi> <mo mathvariant="bold" stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </msub> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {{\bar {X}}_{n}} ={\begin{bmatrix}{\bar {X}}_{i}^{(1)}\\\vdots \\{\bar {X}}_{i}^{(k)}\end{bmatrix}}={\frac {1}{n}}\sum _{i=1}^{n}\mathbf {X} _{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf121f2531614d5deda7f010f4299906d40e7763" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.262ex; margin-bottom: -0.243ex; width:27.525ex; height:12.176ex;" alt="{\displaystyle \mathbf {{\bar {X}}_{n}} ={\begin{bmatrix}{\bar {X}}_{i}^{(1)}\\\vdots \\{\bar {X}}_{i}^{(k)}\end{bmatrix}}={\frac {1}{n}}\sum _{i=1}^{n}\mathbf {X} _{i}.}"></span> Therefore, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {n}}}\sum _{i=1}^{n}\left[\mathbf {X} _{i}-\operatorname {E} \left(\mathbf {X} _{i}\right)\right]={\frac {1}{\sqrt {n}}}\sum _{i=1}^{n}(\mathbf {X} _{i}-{\boldsymbol {\mu }})={\sqrt {n}}\left({\overline {\mathbf {X} }}_{n}-{\boldsymbol {\mu }}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>n</mi> </msqrt> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>[</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>n</mi> </msqrt> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {n}}}\sum _{i=1}^{n}\left[\mathbf {X} _{i}-\operatorname {E} \left(\mathbf {X} _{i}\right)\right]={\frac {1}{\sqrt {n}}}\sum _{i=1}^{n}(\mathbf {X} _{i}-{\boldsymbol {\mu }})={\sqrt {n}}\left({\overline {\mathbf {X} }}_{n}-{\boldsymbol {\mu }}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77d3c26b99d8a97339b688367c519d0e28f9541c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:60.047ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{\sqrt {n}}}\sum _{i=1}^{n}\left[\mathbf {X} _{i}-\operatorname {E} \left(\mathbf {X} _{i}\right)\right]={\frac {1}{\sqrt {n}}}\sum _{i=1}^{n}(\mathbf {X} _{i}-{\boldsymbol {\mu }})={\sqrt {n}}\left({\overline {\mathbf {X} }}_{n}-{\boldsymbol {\mu }}\right).}"></span> </p><p>The multivariate central limit theorem states that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}\left({\overline {\mathbf {X} }}_{n}-{\boldsymbol {\mu }}\right)\mathrel {\overset {d}{\longrightarrow }} {\mathcal {N}}_{k}(0,{\boldsymbol {\Sigma }}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-REL"> <mover> <mo stretchy="false">⟶</mo> <mi>d</mi> </mover> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}}\left({\overline {\mathbf {X} }}_{n}-{\boldsymbol {\mu }}\right)\mathrel {\overset {d}{\longrightarrow }} {\mathcal {N}}_{k}(0,{\boldsymbol {\Sigma }}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e50727443f2f8d94b4e35b66a3d441a12e69e6b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.006ex; height:5.176ex;" alt="{\displaystyle {\sqrt {n}}\left({\overline {\mathbf {X} }}_{n}-{\boldsymbol {\mu }}\right)\mathrel {\overset {d}{\longrightarrow }} {\mathcal {N}}_{k}(0,{\boldsymbol {\Sigma }}),}"></span> where the <a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrix</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\Sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\Sigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8532511177f5a2d2dc2b8c1ea37d483c70266911" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\Sigma }}}"></span> is equal to <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\Sigma }}={\begin{bmatrix}{\operatorname {Var} \left(X_{1}^{(1)}\right)}&\operatorname {Cov} \left(X_{1}^{(1)},X_{1}^{(2)}\right)&\operatorname {Cov} \left(X_{1}^{(1)},X_{1}^{(3)}\right)&\cdots &\operatorname {Cov} \left(X_{1}^{(1)},X_{1}^{(k)}\right)\\\operatorname {Cov} \left(X_{1}^{(2)},X_{1}^{(1)}\right)&\operatorname {Var} \left(X_{1}^{(2)}\right)&\operatorname {Cov} \left(X_{1}^{(2)},X_{1}^{(3)}\right)&\cdots &\operatorname {Cov} \left(X_{1}^{(2)},X_{1}^{(k)}\right)\\\operatorname {Cov} \left(X_{1}^{(3)},X_{1}^{(1)}\right)&\operatorname {Cov} \left(X_{1}^{(3)},X_{1}^{(2)}\right)&\operatorname {Var} \left(X_{1}^{(3)}\right)&\cdots &\operatorname {Cov} \left(X_{1}^{(3)},X_{1}^{(k)}\right)\\\vdots &\vdots &\vdots &\ddots &\vdots \\\operatorname {Cov} \left(X_{1}^{(k)},X_{1}^{(1)}\right)&\operatorname {Cov} \left(X_{1}^{(k)},X_{1}^{(2)}\right)&\operatorname {Cov} \left(X_{1}^{(k)},X_{1}^{(3)}\right)&\cdots &\operatorname {Var} \left(X_{1}^{(k)}\right)\\\end{bmatrix}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Σ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi>Var</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mi>Cov</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi>Cov</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>Cov</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>Cov</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi>Var</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> </mtd> <mtd> <mi>Cov</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>Cov</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>Cov</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi>Cov</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi>Var</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>Cov</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mi>Cov</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi>Cov</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi>Cov</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>Var</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\Sigma }}={\begin{bmatrix}{\operatorname {Var} \left(X_{1}^{(1)}\right)}&\operatorname {Cov} \left(X_{1}^{(1)},X_{1}^{(2)}\right)&\operatorname {Cov} \left(X_{1}^{(1)},X_{1}^{(3)}\right)&\cdots &\operatorname {Cov} \left(X_{1}^{(1)},X_{1}^{(k)}\right)\\\operatorname {Cov} \left(X_{1}^{(2)},X_{1}^{(1)}\right)&\operatorname {Var} \left(X_{1}^{(2)}\right)&\operatorname {Cov} \left(X_{1}^{(2)},X_{1}^{(3)}\right)&\cdots &\operatorname {Cov} \left(X_{1}^{(2)},X_{1}^{(k)}\right)\\\operatorname {Cov} \left(X_{1}^{(3)},X_{1}^{(1)}\right)&\operatorname {Cov} \left(X_{1}^{(3)},X_{1}^{(2)}\right)&\operatorname {Var} \left(X_{1}^{(3)}\right)&\cdots &\operatorname {Cov} \left(X_{1}^{(3)},X_{1}^{(k)}\right)\\\vdots &\vdots &\vdots &\ddots &\vdots \\\operatorname {Cov} \left(X_{1}^{(k)},X_{1}^{(1)}\right)&\operatorname {Cov} \left(X_{1}^{(k)},X_{1}^{(2)}\right)&\operatorname {Cov} \left(X_{1}^{(k)},X_{1}^{(3)}\right)&\cdots &\operatorname {Var} \left(X_{1}^{(k)}\right)\\\end{bmatrix}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6d498478cfaf665725932f6934d0e59d52b8d97" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.838ex; width:88.67ex; height:24.843ex;" alt="{\displaystyle {\boldsymbol {\Sigma }}={\begin{bmatrix}{\operatorname {Var} \left(X_{1}^{(1)}\right)}&\operatorname {Cov} \left(X_{1}^{(1)},X_{1}^{(2)}\right)&\operatorname {Cov} \left(X_{1}^{(1)},X_{1}^{(3)}\right)&\cdots &\operatorname {Cov} \left(X_{1}^{(1)},X_{1}^{(k)}\right)\\\operatorname {Cov} \left(X_{1}^{(2)},X_{1}^{(1)}\right)&\operatorname {Var} \left(X_{1}^{(2)}\right)&\operatorname {Cov} \left(X_{1}^{(2)},X_{1}^{(3)}\right)&\cdots &\operatorname {Cov} \left(X_{1}^{(2)},X_{1}^{(k)}\right)\\\operatorname {Cov} \left(X_{1}^{(3)},X_{1}^{(1)}\right)&\operatorname {Cov} \left(X_{1}^{(3)},X_{1}^{(2)}\right)&\operatorname {Var} \left(X_{1}^{(3)}\right)&\cdots &\operatorname {Cov} \left(X_{1}^{(3)},X_{1}^{(k)}\right)\\\vdots &\vdots &\vdots &\ddots &\vdots \\\operatorname {Cov} \left(X_{1}^{(k)},X_{1}^{(1)}\right)&\operatorname {Cov} \left(X_{1}^{(k)},X_{1}^{(2)}\right)&\operatorname {Cov} \left(X_{1}^{(k)},X_{1}^{(3)}\right)&\cdots &\operatorname {Var} \left(X_{1}^{(k)}\right)\\\end{bmatrix}}~.}"></span> </p><p>The multivariate central limit theorem can be proved using the <a href="/wiki/Cram%C3%A9r%E2%80%93Wold_theorem" title="Cramér–Wold theorem">Cramér–Wold theorem</a>.<sup id="cite_ref-vanderVaart_7-1" class="reference"><a href="#cite_note-vanderVaart-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>The rate of convergence is given by the following <a href="/wiki/Berry%E2%80%93Esseen_theorem" title="Berry–Esseen theorem">Berry–Esseen</a> type result: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></strong><span class="theoreme-tiret"> — </span>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\dots ,X_{n},\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},\dots ,X_{n},\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/036c457524cbc41bf97abfb7f5699e44fefc47f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.057ex; height:2.509ex;" alt="{\displaystyle X_{1},\dots ,X_{n},\dots }"></span> be independent <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a713426956296f1668fce772df3c60b9dde8a685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{d}}"></span>-valued random vectors, each having mean zero. Write <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\sum _{i=1}^{n}X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\sum _{i=1}^{n}X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a560a6c26ccec12fa77776c7c9344e68f82df5af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.064ex; height:6.843ex;" alt="{\displaystyle S=\sum _{i=1}^{n}X_{i}}"></span> and assume <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma =\operatorname {Cov} [S]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> <mo>=</mo> <mi>Cov</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>S</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma =\operatorname {Cov} [S]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bbdd46ff8928b02fc4e37a7fc14c51ceaf58b40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.637ex; height:2.843ex;" alt="{\displaystyle \Sigma =\operatorname {Cov} [S]}"></span> is invertible. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z\sim {\mathcal {N}}(0,\Sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z\sim {\mathcal {N}}(0,\Sigma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a54466e721dd1fc4bc34c972d3fe7ad5b82fb54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.738ex; height:3.009ex;" alt="{\displaystyle Z\sim {\mathcal {N}}(0,\Sigma )}"></span> be a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span>-dimensional Gaussian with the same mean and same covariance matrix as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>. Then for all convex sets <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subseteq \mathbb {R} ^{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>⊆</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\subseteq \mathbb {R} ^{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb1a0be5553eae33cbd69b70e7ae9d849b26d058" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.651ex; height:2.843ex;" alt="{\displaystyle U\subseteq \mathbb {R} ^{d}}"></span>,</span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\mathbb {P} [S\in U]-\mathbb {P} [Z\in U]\right|\leq C\,d^{1/4}\gamma ~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">[</mo> <mi>S</mi> <mo>∈</mo> <mi>U</mi> <mo stretchy="false">]</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">[</mo> <mi>Z</mi> <mo>∈</mo> <mi>U</mi> <mo stretchy="false">]</mo> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mi>C</mi> <mspace width="thinmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <mi>γ</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\mathbb {P} [S\in U]-\mathbb {P} [Z\in U]\right|\leq C\,d^{1/4}\gamma ~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c26c26cb1dedbb0db401fd2ebfb479ec45fb4cc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.646ex; height:3.343ex;" alt="{\displaystyle \left|\mathbb {P} [S\in U]-\mathbb {P} [Z\in U]\right|\leq C\,d^{1/4}\gamma ~,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> is a universal constant, <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =\sum _{i=1}^{n}\operatorname {E} \left[\left\|\Sigma ^{-1/2}X_{i}\right\|_{2}^{3}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msubsup> <mrow> <mo symmetric="true">‖</mo> <mrow> <msup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =\sum _{i=1}^{n}\operatorname {E} \left[\left\|\Sigma ^{-1/2}X_{i}\right\|_{2}^{3}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7904b82b37828200b3a729334fb37932175ef82e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.638ex; height:6.843ex;" alt="{\displaystyle \gamma =\sum _{i=1}^{n}\operatorname {E} \left[\left\|\Sigma ^{-1/2}X_{i}\right\|_{2}^{3}\right]}"></span>,</span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3a8e44a2eb980f856968a6357e3d0a7c22c905f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.058ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|_{2}}"></span> denotes the Euclidean norm on <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a713426956296f1668fce772df3c60b9dde8a685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{d}}"></span>.</span> </p> </div> <p>It is unknown whether the factor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle d^{1/4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle d^{1/4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ed525ca1204085b15a58c7fc9b0e7dd1c38428f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.916ex; height:2.676ex;" alt="{\textstyle d^{1/4}}"></span> is necessary.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="The_Generalized_Central_Limit_Theorem">The Generalized Central Limit Theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=6" title="Edit section: The Generalized Central Limit Theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Generalized Central Limit Theorem (GCLT) was an effort of multiple mathematicians (<a href="/wiki/Sergei_Natanovich_Bernstein" class="mw-redirect" title="Sergei Natanovich Bernstein">Bernstein</a>, <a href="/wiki/Jarl_Waldemar_Lindeberg" title="Jarl Waldemar Lindeberg">Lindeberg</a>, <a href="/wiki/Paul_L%C3%A9vy_(mathematician)" title="Paul Lévy (mathematician)">Lévy</a>, <a href="/wiki/William_Feller" title="William Feller">Feller</a>, <a href="/wiki/Andrey_Kolmogorov" title="Andrey Kolmogorov">Kolmogorov</a>, and others) over the period from 1920 to 1937.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The first published complete proof of the GCLT was in 1937 by <a href="/wiki/Paul_L%C3%A9vy_(mathematician)" title="Paul Lévy (mathematician)">Paul Lévy</a> in French.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> An English language version of the complete proof of the GCLT is available in the translation of <a href="/wiki/Boris_Vladimirovich_Gnedenko" title="Boris Vladimirovich Gnedenko">Gnedenko</a> and <a href="/wiki/Andrey_Kolmogorov" title="Andrey Kolmogorov">Kolmogorov</a>'s 1954 book.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>The statement of the GCLT is as follows:<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><i>A non-degenerate random variable</i> <i>Z</i> <i>is <a href="/wiki/Stable_distribution" title="Stable distribution"><i>α</i>-stable</a> for some</i> 0 < <i>α</i> ≤ 2 <i>if and only if there is an independent, identically distributed sequence of random variables</i> <i>X</i><sub>1</sub>, <i>X</i><sub>2</sub>, <i>X</i><sub>3</sub>, <i>... and constants</i> <i>a</i><sub><i>n</i></sub> > 0, <i>b</i><sub><i>n</i></sub> ∈ ℝ <i>with</i></dd></dl> <dl><dd><dl><dd><i>a</i><sub><i>n</i></sub> (<i>X</i><sub>1</sub> + ... + <i>X</i><sub><i>n</i></sub>) − <i>b</i><sub><i>n</i></sub> → <i>Z</i>.</dd></dl></dd></dl> <dl><dd><i>Here → means the sequence of random variable sums converges in distribution; i.e., the corresponding distributions satisfy</i> <i>F</i><sub><i>n</i></sub>(<i>y</i>) → <i>F</i>(<i>y</i>) <i>at all continuity points of</i> <i>F.</i></dd></dl> <p>In other words, if sums of independent, identically distributed random variables converge in distribution to some <i>Z</i>, then <i>Z</i> must be a <a href="/wiki/Stable_distribution" title="Stable distribution">stable distribution</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Dependent_processes">Dependent processes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=7" title="Edit section: Dependent processes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="CLT_under_weak_dependence">CLT under weak dependence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=8" title="Edit section: CLT under weak dependence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A useful generalization of a sequence of independent, identically distributed random variables is a <a href="/wiki/Mixing_(mathematics)" title="Mixing (mathematics)">mixing</a> random process in discrete time; "mixing" means, roughly, that random variables temporally far apart from one another are nearly independent. Several kinds of mixing are used in ergodic theory and probability theory. See especially <a href="/wiki/Mixing_(mathematics)#Mixing_in_stochastic_processes" title="Mixing (mathematics)">strong mixing</a> (also called α-mixing) defined by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \alpha (n)\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \alpha (n)\to 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4ed99c356c145334ba6ea4de91528e6cc812a00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.468ex; height:2.843ex;" alt="{\textstyle \alpha (n)\to 0}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \alpha (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \alpha (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/468f23bd38dabd879395bf653a352665a569a5e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.692ex; height:2.843ex;" alt="{\textstyle \alpha (n)}"></span> is so-called <a href="/wiki/Mixing_(mathematics)#Mixing_in_stochastic_processes" title="Mixing (mathematics)">strong mixing coefficient</a>. </p><p>A simplified formulation of the central limit theorem under strong mixing is:<sup id="cite_ref-FOOTNOTEBillingsley1995Theorem_27.4_14-0" class="reference"><a href="#cite_note-FOOTNOTEBillingsley1995Theorem_27.4-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem</strong><span class="theoreme-tiret"> — </span>Suppose that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \{X_{1},\ldots ,X_{n},\ldots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \{X_{1},\ldots ,X_{n},\ldots \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8cf831d0f6051f596823e3ab16dec9b32c3605b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.382ex; height:2.843ex;" alt="{\textstyle \{X_{1},\ldots ,X_{n},\ldots \}}"></span> is stationary and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>-mixing with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \alpha _{n}=O\left(n^{-5}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>O</mi> <mrow> <mo>(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>5</mn> </mrow> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \alpha _{n}=O\left(n^{-5}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9101fb0dac273b5c7c43dc1e9477ab1998baca60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.822ex; height:3.176ex;" alt="{\textstyle \alpha _{n}=O\left(n^{-5}\right)}"></span> and that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {E} [X_{n}]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {E} [X_{n}]=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d12315de3945900a1cdcca84088a0f562e93d042" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.28ex; height:2.843ex;" alt="{\textstyle \operatorname {E} [X_{n}]=0}"></span> and <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {E} [X_{n}^{12}]<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msubsup> <mo stretchy="false">]</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {E} [X_{n}^{12}]<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa251162abe2eb1e886116a08905a2e1f30ba891" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.172ex; height:3.009ex;" alt="{\textstyle \operatorname {E} [X_{n}^{12}]<\infty }"></span>.</span> Denote <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle S_{n}=X_{1}+\cdots +X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle S_{n}=X_{1}+\cdots +X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36fd633e84994384183529ab9a056835465b9382" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.267ex; height:2.509ex;" alt="{\textstyle S_{n}=X_{1}+\cdots +X_{n}}"></span>,</span> then the limit <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}=\lim _{n\rightarrow \infty }{\frac {\operatorname {E} \left(S_{n}^{2}\right)}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>)</mo> </mrow> </mrow> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}=\lim _{n\rightarrow \infty }{\frac {\operatorname {E} \left(S_{n}^{2}\right)}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2535aaf054d8ae91ed0f0d2a87929f50bca7eabb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.335ex; height:6.009ex;" alt="{\displaystyle \sigma ^{2}=\lim _{n\rightarrow \infty }{\frac {\operatorname {E} \left(S_{n}^{2}\right)}{n}}}"></span> exists, and if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sigma \neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>σ</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sigma \neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba5fed43f7e3f2424d0be5a899641b1245afaf75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\textstyle \sigma \neq 0}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {S_{n}}{\sigma {\sqrt {n}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {S_{n}}{\sigma {\sqrt {n}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/889c6fe4e873253ac5404d0bbb1909afca5eef22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.131ex; height:4.509ex;" alt="{\textstyle {\frac {S_{n}}{\sigma {\sqrt {n}}}}}"></span> converges in distribution to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {N}}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathcal {N}}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba37ff02211fb81c813065b489b78ec9df951e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:7.505ex; height:3.009ex;" alt="{\textstyle {\mathcal {N}}(0,1)}"></span>. </p> </div> <p>In fact, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}=\operatorname {E} \left(X_{1}^{2}\right)+2\sum _{k=1}^{\infty }\operatorname {E} \left(X_{1}X_{1+k}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}=\operatorname {E} \left(X_{1}^{2}\right)+2\sum _{k=1}^{\infty }\operatorname {E} \left(X_{1}X_{1+k}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb0d3c7756dcb842a098c5ff56495a7e16c4ba3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.51ex; height:6.843ex;" alt="{\displaystyle \sigma ^{2}=\operatorname {E} \left(X_{1}^{2}\right)+2\sum _{k=1}^{\infty }\operatorname {E} \left(X_{1}X_{1+k}\right),}"></span> where the series converges absolutely. </p><p>The assumption <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sigma \neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>σ</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sigma \neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba5fed43f7e3f2424d0be5a899641b1245afaf75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.591ex; height:2.676ex;" alt="{\textstyle \sigma \neq 0}"></span> cannot be omitted, since the asymptotic normality fails for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X_{n}=Y_{n}-Y_{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle X_{n}=Y_{n}-Y_{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f4384d7efbcc99abdb70838f377c73b071b7a59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.32ex; height:2.509ex;" alt="{\textstyle X_{n}=Y_{n}-Y_{n-1}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle Y_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle Y_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f44f731779482cae5f5ef040bfa0cb6c181dd4eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.569ex; height:2.509ex;" alt="{\textstyle Y_{n}}"></span> are another <a href="/wiki/Stationary_sequence" title="Stationary sequence">stationary sequence</a>. </p><p>There is a stronger version of the theorem:<sup id="cite_ref-FOOTNOTEDurrett2004Sect._7.7(c),_Theorem_7.8_15-0" class="reference"><a href="#cite_note-FOOTNOTEDurrett2004Sect._7.7(c),_Theorem_7.8-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> the assumption <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {E} \left[X_{n}^{12}\right]<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msubsup> <mo>]</mo> </mrow> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {E} \left[X_{n}^{12}\right]<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/657baea1ae3ea930c0d5057a69e2353a155a0df4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.817ex; height:3.176ex;" alt="{\textstyle \operatorname {E} \left[X_{n}^{12}\right]<\infty }"></span> is replaced with <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {E} \left[{\left|X_{n}\right|}^{2+\delta }\right]<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mi>δ</mi> </mrow> </msup> <mo>]</mo> </mrow> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {E} \left[{\left|X_{n}\right|}^{2+\delta }\right]<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/940680c6ace3f0b2a962e5e31fcc79b6e4f28f13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.711ex; height:4.843ex;" alt="{\textstyle \operatorname {E} \left[{\left|X_{n}\right|}^{2+\delta }\right]<\infty }"></span>,</span> and the assumption <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \alpha _{n}=O\left(n^{-5}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>O</mi> <mrow> <mo>(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>5</mn> </mrow> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \alpha _{n}=O\left(n^{-5}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9101fb0dac273b5c7c43dc1e9477ab1998baca60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.822ex; height:3.176ex;" alt="{\textstyle \alpha _{n}=O\left(n^{-5}\right)}"></span> is replaced with <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n}\alpha _{n}^{\frac {\delta }{2(2+\delta )}}<\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <msubsup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>δ</mi> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </msubsup> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n}\alpha _{n}^{\frac {\delta }{2(2+\delta )}}<\infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/326a749d6ae5429bb4e90c62fa60214ff2c65d93" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.38ex; height:7.343ex;" alt="{\displaystyle \sum _{n}\alpha _{n}^{\frac {\delta }{2(2+\delta )}}<\infty .}"></span> </p><p>Existence of such <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \delta >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>δ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \delta >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e503a231ad333c0bd9a2dc7d48d8848076423356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\textstyle \delta >0}"></span> ensures the conclusion. For encyclopedic treatment of limit theorems under mixing conditions see (<a href="#CITEREFBradley2007">Bradley 2007</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Martingale_difference_CLT">Martingale difference CLT</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=9" title="Edit section: Martingale difference CLT"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Martingale_central_limit_theorem" title="Martingale central limit theorem">Martingale central limit theorem</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem</strong><span class="theoreme-tiret"> — </span>Let a <a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">martingale</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle M_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle M_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6770260b77798ff7d5e557db457172ec8dacd02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.473ex; height:2.509ex;" alt="{\textstyle M_{n}}"></span> satisfy </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{n}}\sum _{k=1}^{n}\operatorname {E} \left[\left(M_{k}-M_{k-1}\right)^{2}\mid M_{1},\dots ,M_{k-1}\right]\to 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>∣</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo stretchy="false">→</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{n}}\sum _{k=1}^{n}\operatorname {E} \left[\left(M_{k}-M_{k-1}\right)^{2}\mid M_{1},\dots ,M_{k-1}\right]\to 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7fc9c003077c427537a3a787e84ef5528bbe866" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.271ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{n}}\sum _{k=1}^{n}\operatorname {E} \left[\left(M_{k}-M_{k-1}\right)^{2}\mid M_{1},\dots ,M_{k-1}\right]\to 1}"></span> in probability as <span class="texhtml"><i>n</i> → ∞</span>,</li> <li>for every <span class="texhtml"><i>ε</i> > 0</span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{n}}\sum _{k=1}^{n}{\operatorname {E} \left[\left(M_{k}-M_{k-1}\right)^{2}\mathbf {1} \left[|M_{k}-M_{k-1}|>\varepsilon {\sqrt {n}}\right]\right]}\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{n}}\sum _{k=1}^{n}{\operatorname {E} \left[\left(M_{k}-M_{k-1}\right)^{2}\mathbf {1} \left[|M_{k}-M_{k-1}|>\varepsilon {\sqrt {n}}\right]\right]}\to 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3a52a1cd0940afa0a6752d55edc28d1c72ab1bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:52.854ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{n}}\sum _{k=1}^{n}{\operatorname {E} \left[\left(M_{k}-M_{k-1}\right)^{2}\mathbf {1} \left[|M_{k}-M_{k-1}|>\varepsilon {\sqrt {n}}\right]\right]}\to 0}"></span> as <span class="texhtml"><i>n</i> → ∞</span>,</li></ul> <p>then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {M_{n}}{\sqrt {n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msqrt> <mi>n</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {M_{n}}{\sqrt {n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fefdb90dfbb56b1aa437d0dc16f9cab5906c7fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.395ex; height:4.509ex;" alt="{\textstyle {\frac {M_{n}}{\sqrt {n}}}}"></span> converges in distribution to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {N}}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathcal {N}}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba37ff02211fb81c813065b489b78ec9df951e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:7.505ex; height:3.009ex;" alt="{\textstyle {\mathcal {N}}(0,1)}"></span> as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2e2d57fbe15926e75495e28e3517ffec3622d96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\textstyle n\to \infty }"></span>.<sup id="cite_ref-FOOTNOTEDurrett2004Sect._7.7,_Theorem_7.4_16-0" class="reference"><a href="#cite_note-FOOTNOTEDurrett2004Sect._7.7,_Theorem_7.4-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEBillingsley1995Theorem_35.12_17-0" class="reference"><a href="#cite_note-FOOTNOTEBillingsley1995Theorem_35.12-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> </div> <div class="mw-heading mw-heading2"><h2 id="Remarks">Remarks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=10" title="Edit section: Remarks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Proof_of_classical_CLT">Proof of classical CLT</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=11" title="Edit section: Proof of classical CLT"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The central limit theorem has a proof using <a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic functions</a>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> It is similar to the proof of the (weak) <a href="/wiki/Proof_of_the_law_of_large_numbers" class="mw-redirect" title="Proof of the law of large numbers">law of large numbers</a>. </p><p>Assume <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \{X_{1},\ldots ,X_{n},\ldots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \{X_{1},\ldots ,X_{n},\ldots \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8cf831d0f6051f596823e3ab16dec9b32c3605b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.382ex; height:2.843ex;" alt="{\textstyle \{X_{1},\ldots ,X_{n},\ldots \}}"></span> are independent and identically distributed random variables, each with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/259577540a13444806174d5a1ae7662974f58085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\textstyle \mu }"></span> and finite variance <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a86f1d00f664920ef46109bcddc0778f4976b490" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.509ex;" alt="{\textstyle \sigma ^{2}}"></span>.</span> The sum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X_{1}+\cdots +X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle X_{1}+\cdots +X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2dd1d07185beda1344edb462098bdeb9c3d4bd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.525ex; height:2.509ex;" alt="{\textstyle X_{1}+\cdots +X_{n}}"></span> has <a href="/wiki/Linearity_of_expectation" class="mw-redirect" title="Linearity of expectation">mean</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86c2acb561e48611211d932c89b04070975c3097" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.796ex; height:2.176ex;" alt="{\textstyle n\mu }"></span> and <a href="/wiki/Variance#Sum_of_uncorrelated_variables_(Bienaymé_formula)" title="Variance">variance</a> <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n\sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n\sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3ac46172bd48cbd73b45b4cd9866ba79c60c024" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.779ex; height:2.509ex;" alt="{\textstyle n\sigma ^{2}}"></span>.</span> Consider the random variable <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{n}={\frac {X_{1}+\cdots +X_{n}-n\mu }{\sqrt {n\sigma ^{2}}}}=\sum _{i=1}^{n}{\frac {X_{i}-\mu }{\sqrt {n\sigma ^{2}}}}=\sum _{i=1}^{n}{\frac {1}{\sqrt {n}}}Y_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>n</mi> <mi>μ</mi> </mrow> <msqrt> <mi>n</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> </mrow> <msqrt> <mi>n</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>n</mi> </msqrt> </mfrac> </mrow> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{n}={\frac {X_{1}+\cdots +X_{n}-n\mu }{\sqrt {n\sigma ^{2}}}}=\sum _{i=1}^{n}{\frac {X_{i}-\mu }{\sqrt {n\sigma ^{2}}}}=\sum _{i=1}^{n}{\frac {1}{\sqrt {n}}}Y_{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad49250c7f766ddd917c1a211d0a9cc17e22ef07" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:55.35ex; height:6.843ex;" alt="{\displaystyle Z_{n}={\frac {X_{1}+\cdots +X_{n}-n\mu }{\sqrt {n\sigma ^{2}}}}=\sum _{i=1}^{n}{\frac {X_{i}-\mu }{\sqrt {n\sigma ^{2}}}}=\sum _{i=1}^{n}{\frac {1}{\sqrt {n}}}Y_{i},}"></span> where in the last step we defined the new random variables <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle Y_{i}={\frac {X_{i}-\mu }{\sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> </mrow> <mi>σ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle Y_{i}={\frac {X_{i}-\mu }{\sigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d649652d30c90f9eff8ced6aa98a604cce20439" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.34ex; height:3.843ex;" alt="{\textstyle Y_{i}={\frac {X_{i}-\mu }{\sigma }}}"></span>,</span> each with zero mean and unit variance <span class="nowrap">(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {var} (Y)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {var} (Y)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89961aa24aac81f848a2039be849ae38207cbb20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.145ex; height:2.843ex;" alt="{\textstyle \operatorname {var} (Y)=1}"></span>).</span> The <a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle Z_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle Z_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6bb2f366f0fced92341046f7bdc5669190382a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.806ex; height:2.509ex;" alt="{\textstyle Z_{n}}"></span> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{Z_{n}}\!(t)=\varphi _{\sum _{i=1}^{n}{{\frac {1}{\sqrt {n}}}Y_{i}}}\!(t)\ =\ \varphi _{Y_{1}}\!\!\left({\frac {t}{\sqrt {n}}}\right)\varphi _{Y_{2}}\!\!\left({\frac {t}{\sqrt {n}}}\right)\cdots \varphi _{Y_{n}}\!\!\left({\frac {t}{\sqrt {n}}}\right)\ =\ \left[\varphi _{Y_{1}}\!\!\left({\frac {t}{\sqrt {n}}}\right)\right]^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mspace width="negativethinmathspace" /> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>n</mi> </msqrt> </mfrac> </mrow> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mrow> </msub> <mspace width="negativethinmathspace" /> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <msqrt> <mi>n</mi> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <msqrt> <mi>n</mi> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>⋯</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <msqrt> <mi>n</mi> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <msup> <mrow> <mo>[</mo> <mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <msqrt> <mi>n</mi> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{Z_{n}}\!(t)=\varphi _{\sum _{i=1}^{n}{{\frac {1}{\sqrt {n}}}Y_{i}}}\!(t)\ =\ \varphi _{Y_{1}}\!\!\left({\frac {t}{\sqrt {n}}}\right)\varphi _{Y_{2}}\!\!\left({\frac {t}{\sqrt {n}}}\right)\cdots \varphi _{Y_{n}}\!\!\left({\frac {t}{\sqrt {n}}}\right)\ =\ \left[\varphi _{Y_{1}}\!\!\left({\frac {t}{\sqrt {n}}}\right)\right]^{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412edc029b680d1254078fd2c5727db6cf30ea76" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:82.199ex; height:6.509ex;" alt="{\displaystyle \varphi _{Z_{n}}\!(t)=\varphi _{\sum _{i=1}^{n}{{\frac {1}{\sqrt {n}}}Y_{i}}}\!(t)\ =\ \varphi _{Y_{1}}\!\!\left({\frac {t}{\sqrt {n}}}\right)\varphi _{Y_{2}}\!\!\left({\frac {t}{\sqrt {n}}}\right)\cdots \varphi _{Y_{n}}\!\!\left({\frac {t}{\sqrt {n}}}\right)\ =\ \left[\varphi _{Y_{1}}\!\!\left({\frac {t}{\sqrt {n}}}\right)\right]^{n},}"></span> where in the last step we used the fact that all of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle Y_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle Y_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b48694270212208c6f289fd3f5cd0c85a2ab1426" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.15ex; height:2.509ex;" alt="{\textstyle Y_{i}}"></span> are identically distributed. The characteristic function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle Y_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle Y_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d99286ab1964199302400d49e2efbacff8e0df7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.405ex; height:2.509ex;" alt="{\textstyle Y_{1}}"></span> is, by <a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{Y_{1}}\!\left({\frac {t}{\sqrt {n}}}\right)=1-{\frac {t^{2}}{2n}}+o\!\left({\frac {t^{2}}{n}}\right),\quad \left({\frac {t}{\sqrt {n}}}\right)\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <msqrt> <mi>n</mi> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>o</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <msqrt> <mi>n</mi> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{Y_{1}}\!\left({\frac {t}{\sqrt {n}}}\right)=1-{\frac {t^{2}}{2n}}+o\!\left({\frac {t^{2}}{n}}\right),\quad \left({\frac {t}{\sqrt {n}}}\right)\to 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef0090e6f84b8f85bbd892cdb5b6d12e7e05c3e0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:47.849ex; height:6.676ex;" alt="{\displaystyle \varphi _{Y_{1}}\!\left({\frac {t}{\sqrt {n}}}\right)=1-{\frac {t^{2}}{2n}}+o\!\left({\frac {t^{2}}{n}}\right),\quad \left({\frac {t}{\sqrt {n}}}\right)\to 0}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle o(t^{2}/n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>o</mi> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle o(t^{2}/n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/532efbd52ebff5b1011012e68aed287fd957abc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.388ex; height:3.009ex;" alt="{\textstyle o(t^{2}/n)}"></span> is "<a href="/wiki/Little-o_notation" class="mw-redirect" title="Little-o notation">little <span class="texhtml mvar" style="font-style:italic;">o</span> notation</a>" for some function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2bc926f90178739fccd01a96c6fa778ab3535d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\textstyle t}"></span> that goes to zero more rapidly than <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle t^{2}/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle t^{2}/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3697c7a2d39c44bf73f66827f536ff885c0d92b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.451ex; height:3.009ex;" alt="{\textstyle t^{2}/n}"></span>.</span> By the limit of the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a> <span class="nowrap">(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle e^{x}=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle e^{x}=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ea7045a332544d742dd5772aca36c5a4ea64b36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.649ex; height:3.343ex;" alt="{\textstyle e^{x}=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}}"></span>),</span> the characteristic function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5073995dface6fb94824a8bec0075e65205fc64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.806ex; height:2.509ex;" alt="{\displaystyle Z_{n}}"></span> equals <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{Z_{n}}(t)=\left(1-{\frac {t^{2}}{2n}}+o\left({\frac {t^{2}}{n}}\right)\right)^{n}\rightarrow e^{-{\frac {1}{2}}t^{2}},\quad n\to \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>o</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{Z_{n}}(t)=\left(1-{\frac {t^{2}}{2n}}+o\left({\frac {t^{2}}{n}}\right)\right)^{n}\rightarrow e^{-{\frac {1}{2}}t^{2}},\quad n\to \infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f99cfa0ea68282331a0ce987cab1c839a055bf9d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:52.602ex; height:6.343ex;" alt="{\displaystyle \varphi _{Z_{n}}(t)=\left(1-{\frac {t^{2}}{2n}}+o\left({\frac {t^{2}}{n}}\right)\right)^{n}\rightarrow e^{-{\frac {1}{2}}t^{2}},\quad n\to \infty .}"></span> </p><p>All of the higher order terms vanish in the limit <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2e2d57fbe15926e75495e28e3517ffec3622d96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\textstyle n\to \infty }"></span>.</span> The right hand side equals the characteristic function of a standard normal distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {N}}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathcal {N}}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba37ff02211fb81c813065b489b78ec9df951e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:7.505ex; height:3.009ex;" alt="{\textstyle {\mathcal {N}}(0,1)}"></span>, which implies through <a href="/wiki/L%C3%A9vy_continuity_theorem" class="mw-redirect" title="Lévy continuity theorem">Lévy's continuity theorem</a> that the distribution of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle Z_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle Z_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6bb2f366f0fced92341046f7bdc5669190382a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.806ex; height:2.509ex;" alt="{\textstyle Z_{n}}"></span> will approach <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {N}}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathcal {N}}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba37ff02211fb81c813065b489b78ec9df951e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:7.505ex; height:3.009ex;" alt="{\textstyle {\mathcal {N}}(0,1)}"></span> as <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2e2d57fbe15926e75495e28e3517ffec3622d96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\textstyle n\to \infty }"></span>.</span> Therefore, the <a href="/wiki/Sample_mean" class="mw-redirect" title="Sample mean">sample average</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {X}}_{n}={\frac {X_{1}+\cdots +X_{n}}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {X}}_{n}={\frac {X_{1}+\cdots +X_{n}}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9e4899d5168356e3bfce210ef39b7a67326d613" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.658ex; height:5.176ex;" alt="{\displaystyle {\bar {X}}_{n}={\frac {X_{1}+\cdots +X_{n}}{n}}}"></span> is such that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sqrt {n}}{\sigma }}({\bar {X}}_{n}-\mu )=Z_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mi>n</mi> </msqrt> <mi>σ</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {n}}{\sigma }}({\bar {X}}_{n}-\mu )=Z_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33289308d622f37ab721b66cca7b47d55f6e0b51" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.321ex; height:5.843ex;" alt="{\displaystyle {\frac {\sqrt {n}}{\sigma }}({\bar {X}}_{n}-\mu )=Z_{n}}"></span> converges to the normal distribution <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {N}}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathcal {N}}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba37ff02211fb81c813065b489b78ec9df951e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:7.505ex; height:3.009ex;" alt="{\textstyle {\mathcal {N}}(0,1)}"></span>,</span> from which the central limit theorem follows. </p> <div class="mw-heading mw-heading3"><h3 id="Convergence_to_the_limit">Convergence to the limit</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=12" title="Edit section: Convergence to the limit"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The central limit theorem gives only an <a href="/wiki/Asymptotic_distribution" title="Asymptotic distribution">asymptotic distribution</a>. As an approximation for a finite number of observations, it provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="Not immediately obvious, I didn't find a source via google (July 2016)">citation needed</span></a></i>]</sup> </p><p>The convergence in the central limit theorem is <a href="/wiki/Uniform_convergence" title="Uniform convergence">uniform</a> because the limiting cumulative distribution function is continuous. If the third central <a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">moment</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {E} \left[(X_{1}-\mu )^{3}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {E} \left[(X_{1}-\mu )^{3}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/389b355cd56db15cdf7e88c8b0aff830a381726f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.606ex; height:3.176ex;" alt="{\textstyle \operatorname {E} \left[(X_{1}-\mu )^{3}\right]}"></span> exists and is finite, then the speed of convergence is at least on the order of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 1/{\sqrt {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 1/{\sqrt {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7ffe079afd996740883579f310cedf037846363" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.656ex; height:2.843ex;" alt="{\textstyle 1/{\sqrt {n}}}"></span> (see <a href="/wiki/Berry%E2%80%93Esseen_theorem" title="Berry–Esseen theorem">Berry–Esseen theorem</a>). <a href="/wiki/Stein%27s_method" title="Stein's method">Stein's method</a><sup id="cite_ref-stein1972_19-0" class="reference"><a href="#cite_note-stein1972-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> can be used not only to prove the central limit theorem, but also to provide bounds on the rates of convergence for selected metrics.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>The convergence to the normal distribution is monotonic, in the sense that the <a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">entropy</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle Z_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle Z_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6bb2f366f0fced92341046f7bdc5669190382a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.806ex; height:2.509ex;" alt="{\textstyle Z_{n}}"></span> increases <a href="/wiki/Monotonic_function" title="Monotonic function">monotonically</a> to that of the normal distribution.<sup id="cite_ref-ABBN_21-0" class="reference"><a href="#cite_note-ABBN-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>The central limit theorem applies in particular to sums of independent and identically distributed <a href="/wiki/Discrete_random_variable" class="mw-redirect" title="Discrete random variable">discrete random variables</a>. A sum of <a href="/wiki/Discrete_random_variable" class="mw-redirect" title="Discrete random variable">discrete random variables</a> is still a <a href="/wiki/Discrete_random_variable" class="mw-redirect" title="Discrete random variable">discrete random variable</a>, so that we are confronted with a sequence of <a href="/wiki/Discrete_random_variable" class="mw-redirect" title="Discrete random variable">discrete random variables</a> whose cumulative probability distribution function converges towards a cumulative probability distribution function corresponding to a continuous variable (namely that of the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>). This means that if we build a <a href="/wiki/Histogram" title="Histogram">histogram</a> of the realizations of the sum of <span class="texhtml mvar" style="font-style:italic;">n</span> independent identical discrete variables, the piecewise-linear curve that joins the centers of the upper faces of the rectangles forming the histogram converges toward a Gaussian curve as <span class="texhtml mvar" style="font-style:italic;">n</span> approaches infinity; this relation is known as <a href="/wiki/De_Moivre%E2%80%93Laplace_theorem" title="De Moivre–Laplace theorem">de Moivre–Laplace theorem</a>. The <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a> article details such an application of the central limit theorem in the simple case of a discrete variable taking only two possible values. </p> <div class="mw-heading mw-heading3"><h3 id="Common_misconceptions">Common misconceptions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=13" title="Edit section: Common misconceptions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Studies have shown that the central limit theorem is subject to several common but serious misconceptions, some of which appear in widely used textbooks.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> These include: </p> <ul><li>The misconceived belief that the theorem applies to random sampling of any variable, rather than to the mean values (or sums) of <a href="/wiki/Iid" class="mw-redirect" title="Iid">iid</a> random variables extracted from a population by repeated sampling. That is, the theorem assumes the random sampling produces a sampling distribution formed from different values of means (or sums) of such random variables.</li> <li>The misconceived belief that the theorem ensures that random sampling leads to the emergence of a normal distribution for sufficiently large samples of any random variable, regardless of the population distribution. In reality, such sampling asymptotically reproduces the properties of the population, an intuitive result underpinned by the <a href="/wiki/Glivenko-Cantelli_theorem" class="mw-redirect" title="Glivenko-Cantelli theorem">Glivenko-Cantelli theorem</a>.</li> <li>The misconceived belief that the theorem leads to a good approximation of a normal distribution for sample sizes greater than around 30,<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> allowing reliable inferences regardless of the nature of the population. In reality, this empirical rule of thumb has no valid justification, and can lead to seriously flawed inferences. See <a href="/wiki/Z-test" title="Z-test">Z-test</a> for where the approximation holds.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Relation_to_the_law_of_large_numbers">Relation to the law of large numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=14" title="Edit section: Relation to the law of large numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Law_of_large_numbers" title="Law of large numbers">law of large numbers</a> as well as the central limit theorem are partial solutions to a general problem: "What is the limiting behavior of <span class="texhtml">S<sub><span class="texhtml mvar" style="font-style:italic;">n</span></sub></span> as <span class="texhtml mvar" style="font-style:italic;">n</span> approaches infinity?" In mathematical analysis, <a href="/wiki/Asymptotic_series" class="mw-redirect" title="Asymptotic series">asymptotic series</a> are one of the most popular tools employed to approach such questions. </p><p>Suppose we have an asymptotic expansion of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f435bf426ab479f25120bda0b15bb6f1f3ddab4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="{\textstyle f(n)}"></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)=a_{1}\varphi _{1}(n)+a_{2}\varphi _{2}(n)+O{\big (}\varphi _{3}(n){\big )}\qquad (n\to \infty ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)=a_{1}\varphi _{1}(n)+a_{2}\varphi _{2}(n)+O{\big (}\varphi _{3}(n){\big )}\qquad (n\to \infty ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/260a39ef95ac12c7c49207851f142e04231b3bbf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:53.502ex; height:3.176ex;" alt="{\displaystyle f(n)=a_{1}\varphi _{1}(n)+a_{2}\varphi _{2}(n)+O{\big (}\varphi _{3}(n){\big )}\qquad (n\to \infty ).}"></span> </p><p>Dividing both parts by <span class="texhtml"><i>φ</i><sub>1</sub>(<i>n</i>)</span> and taking the limit will produce <span class="texhtml"><i>a</i><sub>1</sub></span>, the coefficient of the highest-order term in the expansion, which represents the rate at which <span class="texhtml"><i>f</i>(<i>n</i>)</span> changes in its leading term. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{\varphi _{1}(n)}}=a_{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{\varphi _{1}(n)}}=a_{1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44cc891b6ed15c5597004a713bb9254f19e922a6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.303ex; height:6.509ex;" alt="{\displaystyle \lim _{n\to \infty }{\frac {f(n)}{\varphi _{1}(n)}}=a_{1}.}"></span> </p><p>Informally, one can say: "<span class="texhtml"><i>f</i>(<i>n</i>)</span> grows approximately as <span class="texhtml"><i>a</i><sub>1</sub><i>φ</i><sub>1</sub>(<i>n</i>)</span>". Taking the difference between <span class="texhtml"><i>f</i>(<i>n</i>)</span> and its approximation and then dividing by the next term in the expansion, we arrive at a more refined statement about <span class="texhtml"><i>f</i>(<i>n</i>)</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{\frac {f(n)-a_{1}\varphi _{1}(n)}{\varphi _{2}(n)}}=a_{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {f(n)-a_{1}\varphi _{1}(n)}{\varphi _{2}(n)}}=a_{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fced405a36296ad3d95ea743491cf3773124a0b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.911ex; height:6.509ex;" alt="{\displaystyle \lim _{n\to \infty }{\frac {f(n)-a_{1}\varphi _{1}(n)}{\varphi _{2}(n)}}=a_{2}.}"></span> </p><p>Here one can say that the difference between the function and its approximation grows approximately as <span class="texhtml"><i>a</i><sub>2</sub><i>φ</i><sub>2</sub>(<i>n</i>)</span>. The idea is that dividing the function by appropriate normalizing functions, and looking at the limiting behavior of the result, can tell us much about the limiting behavior of the original function itself. </p><p>Informally, something along these lines happens when the sum, <span class="texhtml mvar" style="font-style:italic;">S<sub>n</sub></span>, of independent identically distributed random variables, <span class="texhtml"><i>X</i><sub>1</sub>, ..., <i>X<sub>n</sub></i></span>, is studied in classical probability theory.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2012)">citation needed</span></a></i>]</sup> If each <span class="texhtml mvar" style="font-style:italic;">X<sub>i</sub></span> has finite mean <span class="texhtml mvar" style="font-style:italic;">μ</span>, then by the law of large numbers, <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>S<sub>n</sub></i></span><span class="sr-only">/</span><span class="den"><i>n</i></span></span>⁠</span> → <i>μ</i></span>.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> If in addition each <span class="texhtml mvar" style="font-style:italic;">X<sub>i</sub></span> has finite variance <span class="texhtml"><i>σ</i><sup>2</sup></span>, then by the central limit theorem, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {S_{n}-n\mu }{\sqrt {n}}}\to \xi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>n</mi> <mi>μ</mi> </mrow> <msqrt> <mi>n</mi> </msqrt> </mfrac> </mrow> <mo stretchy="false">→</mo> <mi>ξ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {S_{n}-n\mu }{\sqrt {n}}}\to \xi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab93c204de66a8de1792dce791a455d187e2899" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.407ex; height:6.509ex;" alt="{\displaystyle {\frac {S_{n}-n\mu }{\sqrt {n}}}\to \xi ,}"></span> where <span class="texhtml mvar" style="font-style:italic;">ξ</span> is distributed as <span class="texhtml"><i>N</i>(0,<i>σ</i><sup>2</sup>)</span>. This provides values of the first two constants in the informal expansion <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}\approx \mu n+\xi {\sqrt {n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≈</mo> <mi>μ</mi> <mi>n</mi> <mo>+</mo> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}\approx \mu n+\xi {\sqrt {n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d486bc5083bebdb5d386be40624936c849610b3c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.386ex; height:3.009ex;" alt="{\displaystyle S_{n}\approx \mu n+\xi {\sqrt {n}}.}"></span> </p><p>In the case where the <span class="texhtml mvar" style="font-style:italic;">X<sub>i</sub></span> do not have finite mean or variance, convergence of the shifted and rescaled sum can also occur with different centering and scaling factors: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {S_{n}-a_{n}}{b_{n}}}\rightarrow \Xi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo stretchy="false">→</mo> <mi mathvariant="normal">Ξ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {S_{n}-a_{n}}{b_{n}}}\rightarrow \Xi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d90d434da533ea39b8411d6cb5bd25a0d2f6725" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.579ex; height:5.843ex;" alt="{\displaystyle {\frac {S_{n}-a_{n}}{b_{n}}}\rightarrow \Xi ,}"></span> or informally <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}\approx a_{n}+\Xi b_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≈</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Ξ</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}\approx a_{n}+\Xi b_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3defe7d02c1f798ba4cb00acf610edbd79fe573" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.444ex; height:2.509ex;" alt="{\displaystyle S_{n}\approx a_{n}+\Xi b_{n}.}"></span> </p><p>Distributions <span class="texhtml">Ξ</span> which can arise in this way are called <i><a href="/wiki/Stable_distribution" title="Stable distribution">stable</a></i>.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> Clearly, the normal distribution is stable, but there are also other stable distributions, such as the <a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy distribution</a>, for which the mean or variance are not defined. The scaling factor <span class="texhtml mvar" style="font-style:italic;">b<sub>n</sub></span> may be proportional to <span class="texhtml mvar" style="font-style:italic;">n<sup>c</sup></span>, for any <span class="texhtml"><i>c</i> ≥ <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>; it may also be multiplied by a <a href="/wiki/Slowly_varying_function" title="Slowly varying function">slowly varying function</a> of <span class="texhtml mvar" style="font-style:italic;">n</span>.<sup id="cite_ref-Uchaikin_28-0" class="reference"><a href="#cite_note-Uchaikin-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Law_of_the_iterated_logarithm" title="Law of the iterated logarithm">law of the iterated logarithm</a> specifies what is happening "in between" the <a href="/wiki/Law_of_large_numbers" title="Law of large numbers">law of large numbers</a> and the central limit theorem. Specifically it says that the normalizing function <span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>n</i> log log <i>n</i></span></span></span>, intermediate in size between <span class="texhtml mvar" style="font-style:italic;">n</span> of the law of large numbers and <span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>n</i></span></span></span> of the central limit theorem, provides a non-trivial limiting behavior. </p> <div class="mw-heading mw-heading3"><h3 id="Alternative_statements_of_the_theorem">Alternative statements of the theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=15" title="Edit section: Alternative statements of the theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Density_functions">Density functions</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=16" title="Edit section: Density functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Probability_density_function" title="Probability density function">density</a> of the sum of two or more independent variables is the <a href="/wiki/Convolution" title="Convolution">convolution</a> of their densities (if these densities exist). Thus the central limit theorem can be interpreted as a statement about the properties of density functions under convolution: the convolution of a number of density functions tends to the normal density as the number of density functions increases without bound. These theorems require stronger hypotheses than the forms of the central limit theorem given above. Theorems of this type are often called local limit theorems. See Petrov<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> for a particular local limit theorem for sums of <a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">independent and identically distributed random variables</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Characteristic_functions">Characteristic functions</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=17" title="Edit section: Characteristic functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since the <a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a> of a convolution is the product of the characteristic functions of the densities involved, the central limit theorem has yet another restatement: the product of the characteristic functions of a number of density functions becomes close to the characteristic function of the normal density as the number of density functions increases without bound, under the conditions stated above. Specifically, an appropriate scaling factor needs to be applied to the argument of the characteristic function. </p><p>An equivalent statement can be made about <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transforms</a>, since the characteristic function is essentially a Fourier transform. </p> <div class="mw-heading mw-heading3"><h3 id="Calculating_the_variance">Calculating the variance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=18" title="Edit section: Calculating the variance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml mvar" style="font-style:italic;">S<sub>n</sub></span> be the sum of <span class="texhtml mvar" style="font-style:italic;">n</span> random variables. Many central limit theorems provide conditions such that <span class="texhtml"><span class="texhtml mvar" style="font-style:italic;">S<sub>n</sub></span>/<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">Var(<span class="texhtml mvar" style="font-style:italic;">S<sub>n</sub></span>)</span></span></span> converges in distribution to <span class="texhtml"><i>N</i>(0,1)</span> (the normal distribution with mean 0, variance 1) as <span class="texhtml"><span class="texhtml mvar" style="font-style:italic;">n</span> → ∞</span>. In some cases, it is possible to find a constant <span class="texhtml"><i>σ</i><sup>2</sup></span> and function <span class="texhtml mvar" style="font-style:italic;">f(n)</span> such that <span class="texhtml"><span class="texhtml mvar" style="font-style:italic;">S<sub>n</sub></span>/(σ<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><span class="texhtml mvar" style="font-style:italic;">n⋅f</span>(<span class="texhtml mvar" style="font-style:italic;">n</span>)</span></span>)</span> converges in distribution to <span class="texhtml"><i>N</i>(0,1)</span> as <span class="texhtml"><span class="texhtml mvar" style="font-style:italic;">n</span>→ ∞</span>. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> <p><strong class="theorem-name">Lemma<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup></strong><span class="theoreme-tiret"> — </span>Suppose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d284b1a1ab85cfbe91d035c6a3bc7a1d17cd31a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.748ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},\dots }"></span> is a sequence of real-valued and strictly stationary random variables with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} (X_{i})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (X_{i})=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e3b337f629495e7455d1c7eba3914a53ef97f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.377ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} (X_{i})=0}"></span> for all <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>,</span> <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:[0,1]\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g:[0,1]\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26aef45c20ce13e8d53e79e068df9b5804c5c170" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.998ex; height:2.843ex;" alt="{\displaystyle g:[0,1]\to \mathbb {R} }"></span>,</span> and <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}=\sum _{i=1}^{n}g\left({\tfrac {i}{n}}\right)X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>g</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>i</mi> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}=\sum _{i=1}^{n}g\left({\tfrac {i}{n}}\right)X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e2b72930e6a989d55d9e95d1509f01e34ffebf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.05ex; height:6.843ex;" alt="{\displaystyle S_{n}=\sum _{i=1}^{n}g\left({\tfrac {i}{n}}\right)X_{i}}"></span>.</span> Construct <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}=\operatorname {E} (X_{1}^{2})+2\sum _{i=1}^{\infty }\operatorname {E} (X_{1}X_{1+i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}=\operatorname {E} (X_{1}^{2})+2\sum _{i=1}^{\infty }\operatorname {E} (X_{1}X_{1+i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f955c402c4eef319d7b8b4af89a317822359d9ca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.254ex; height:6.843ex;" alt="{\displaystyle \sigma ^{2}=\operatorname {E} (X_{1}^{2})+2\sum _{i=1}^{\infty }\operatorname {E} (X_{1}X_{1+i})}"></span> </p> <ol><li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{\infty }\operatorname {E} (X_{1}X_{1+i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{\infty }\operatorname {E} (X_{1}X_{1+i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11f66d88dd1515d176b4906f98ee72938695bce8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.937ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{\infty }\operatorname {E} (X_{1}X_{1+i})}"></span> is absolutely convergent, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\int _{0}^{1}g(x)g'(x)\,dx\right|<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\int _{0}^{1}g(x)g'(x)\,dx\right|<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff2cd32256310d51ceb91bc120c3a0ca535027f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.751ex; height:6.509ex;" alt="{\displaystyle \left|\int _{0}^{1}g(x)g'(x)\,dx\right|<\infty }"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<\int _{0}^{1}(g(x))^{2}dx<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>x</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<\int _{0}^{1}(g(x))^{2}dx<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adf443a91a2107307b730cbc39d29a3a8995f3eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.866ex; height:6.176ex;" alt="{\displaystyle 0<\int _{0}^{1}(g(x))^{2}dx<\infty }"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Var} (S_{n})/(n\gamma _{n})\to \sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Var} (S_{n})/(n\gamma _{n})\to \sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96ca5fa3ea9a2151545d586041dcdbf816e5f8e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.058ex; height:3.176ex;" alt="{\displaystyle \mathrm {Var} (S_{n})/(n\gamma _{n})\to \sigma ^{2}}"></span> as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\to \infty }"></span> where <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{n}={\frac {1}{n}}\sum _{i=1}^{n}\left(g\left({\tfrac {i}{n}}\right)\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow> <mi>g</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>i</mi> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{n}={\frac {1}{n}}\sum _{i=1}^{n}\left(g\left({\tfrac {i}{n}}\right)\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5838491977c539d44aa25f92fcfa8a165cf24ff2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.521ex; height:6.843ex;" alt="{\displaystyle \gamma _{n}={\frac {1}{n}}\sum _{i=1}^{n}\left(g\left({\tfrac {i}{n}}\right)\right)^{2}}"></span>.</span></li> <li>If in addition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/762ecd0f0905dd0d4d7a07f80fa8bfb324b9b021" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle \sigma >0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}/{\sqrt {\mathrm {Var} (S_{n})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}/{\sqrt {\mathrm {Var} (S_{n})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0fa1657c056ad1bec10401d2b948e72c3501e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.399ex; height:4.843ex;" alt="{\displaystyle S_{n}/{\sqrt {\mathrm {Var} (S_{n})}}}"></span> converges in distribution to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3eeb356405c0b33b680b5caa425ada4e9f53e8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:7.505ex; height:3.009ex;" alt="{\displaystyle {\mathcal {N}}(0,1)}"></span> as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\to \infty }"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}/(\sigma {\sqrt {n\gamma _{n}}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}/(\sigma {\sqrt {n\gamma _{n}}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cffedd221b40182d9d29e2250061e39338b60ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.698ex; height:3.176ex;" alt="{\displaystyle S_{n}/(\sigma {\sqrt {n\gamma _{n}}})}"></span> also converges in distribution to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3eeb356405c0b33b680b5caa425ada4e9f53e8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:7.505ex; height:3.009ex;" alt="{\displaystyle {\mathcal {N}}(0,1)}"></span> as <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\to \infty }"></span>.</span></li></ol> </div> <div class="mw-heading mw-heading2"><h2 id="Extensions">Extensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=19" title="Edit section: Extensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Products_of_positive_random_variables">Products of positive random variables</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=20" title="Edit section: Products of positive random variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Logarithm" title="Logarithm">logarithm</a> of a product is simply the sum of the logarithms of the factors. Therefore, when the logarithm of a product of random variables that take only positive values approaches a normal distribution, the product itself approaches a <a href="/wiki/Log-normal_distribution" title="Log-normal distribution">log-normal distribution</a>. Many physical quantities (especially mass or length, which are a matter of scale and cannot be negative) are the products of different <a href="/wiki/Random" class="mw-redirect" title="Random">random</a> factors, so they follow a log-normal distribution. This multiplicative version of the central limit theorem is sometimes called <a href="/wiki/Gibrat%27s_law" title="Gibrat's law">Gibrat's law</a>. </p><p>Whereas the central limit theorem for sums of random variables requires the condition of finite variance, the corresponding theorem for products requires the corresponding condition that the density function be square-integrable.<sup id="cite_ref-Rempala_32-0" class="reference"><a href="#cite_note-Rempala-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Beyond_the_classical_framework">Beyond the classical framework</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=21" title="Edit section: Beyond the classical framework"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Asymptotic normality, that is, <a href="/wiki/Convergence_in_distribution" class="mw-redirect" title="Convergence in distribution">convergence</a> to the normal distribution after appropriate shift and rescaling, is a phenomenon much more general than the classical framework treated above, namely, sums of independent random variables (or vectors). New frameworks are revealed from time to time; no single unifying framework is available for now. </p> <div class="mw-heading mw-heading3"><h3 id="Convex_body">Convex body</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=22" title="Edit section: Convex body"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem</strong><span class="theoreme-tiret"> — </span>There exists a sequence <span class="texhtml"><i>ε<sub>n</sub></i> ↓ 0</span> for which the following holds. Let <span class="texhtml"><i>n</i> ≥ 1</span>, and let random variables <span class="texhtml"><i>X</i><sub>1</sub>, ..., <i>X<sub>n</sub></i></span> have a <a href="/wiki/Logarithmically_concave_function" title="Logarithmically concave function">log-concave</a> <a href="/wiki/Joint_density_function" class="mw-redirect" title="Joint density function">joint density</a> <span class="texhtml mvar" style="font-style:italic;">f</span> such that <span class="texhtml"><i>f</i>(<i>x</i><sub>1</sub>, ..., <i>x<sub>n</sub></i>) = <i>f</i>(|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x</i><sub>1</sub></span>|, ..., |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x<sub>n</sub></i></span>|)</span> for all <span class="texhtml"><i>x</i><sub>1</sub>, ..., <i>x<sub>n</sub></i></span>, and <span class="texhtml">E(<i>X</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>k</i></sub></span></span>) = 1</span> for all <span class="texhtml"><i>k</i> = 1, ..., <i>n</i></span>. Then the distribution of <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {X_{1}+\cdots +X_{n}}{\sqrt {n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <msqrt> <mi>n</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {X_{1}+\cdots +X_{n}}{\sqrt {n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/506e5cbbd5ebf53c483a6b0434c76bca48a9bdfb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:15.361ex; height:6.176ex;" alt="{\displaystyle {\frac {X_{1}+\cdots +X_{n}}{\sqrt {n}}}}"></span> is <span class="texhtml mvar" style="font-style:italic;">ε<sub>n</sub></span>-close to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {N}}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathcal {N}}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba37ff02211fb81c813065b489b78ec9df951e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:7.505ex; height:3.009ex;" alt="{\textstyle {\mathcal {N}}(0,1)}"></span> in the <a href="/wiki/Total_variation_distance_of_probability_measures" title="Total variation distance of probability measures">total variation distance</a>.<sup id="cite_ref-FOOTNOTEKlartag2007Theorem_1.2_33-0" class="reference"><a href="#cite_note-FOOTNOTEKlartag2007Theorem_1.2-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p> </div> <p>These two <span class="texhtml mvar" style="font-style:italic;">ε<sub>n</sub></span>-close distributions have densities (in fact, log-concave densities), thus, the total variance distance between them is the integral of the absolute value of the difference between the densities. Convergence in total variation is stronger than weak convergence. </p><p>An important example of a log-concave density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains the term "central limit theorem for convex bodies". </p><p>Another example: <span class="texhtml"><i>f</i>(<i>x</i><sub>1</sub>, ..., <i>x<sub>n</sub></i>) = const · exp(−(|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x</i><sub>1</sub></span>|<sup><i>α</i></sup> + ⋯ + |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x<sub>n</sub></i></span>|<sup><i>α</i></sup>)<sup><i>β</i></sup>)</span> where <span class="texhtml"><i>α</i> > 1</span> and <span class="texhtml"><i>αβ</i> > 1</span>. If <span class="texhtml"><i>β</i> = 1</span> then <span class="texhtml"><i>f</i>(<i>x</i><sub>1</sub>, ..., <i>x<sub>n</sub></i>)</span> factorizes into <span class="texhtml">const · exp (−|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x</i><sub>1</sub></span>|<sup><i>α</i></sup>) … exp(−|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x<sub>n</sub></i></span>|<sup><i>α</i></sup>), </span> which means <span class="texhtml"><i>X</i><sub>1</sub>, ..., <i>X<sub>n</sub></i></span> are independent. In general, however, they are dependent. </p><p>The condition <span class="texhtml"><i>f</i>(<i>x</i><sub>1</sub>, ..., <i>x<sub>n</sub></i>) = <i>f</i>(|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x</i><sub>1</sub></span>|, ..., |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x<sub>n</sub></i></span>|)</span> ensures that <span class="texhtml"><i>X</i><sub>1</sub>, ..., <i>X<sub>n</sub></i></span> are of zero mean and <a href="/wiki/Uncorrelated" class="mw-redirect" title="Uncorrelated">uncorrelated</a>;<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (June 2012)">citation needed</span></a></i>]</sup> still, they need not be independent, nor even <a href="/wiki/Pairwise_independence" title="Pairwise independence">pairwise independent</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (June 2012)">citation needed</span></a></i>]</sup> By the way, pairwise independence cannot replace independence in the classical central limit theorem.<sup id="cite_ref-FOOTNOTEDurrett2004Section_2.4,_Example_4.5_34-0" class="reference"><a href="#cite_note-FOOTNOTEDurrett2004Section_2.4,_Example_4.5-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p><p>Here is a <a href="/wiki/Berry%E2%80%93Esseen_theorem" title="Berry–Esseen theorem">Berry–Esseen</a> type result. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem</strong><span class="theoreme-tiret"> — </span>Let <span class="texhtml"><i>X</i><sub>1</sub>, ..., <i>X<sub>n</sub></i></span> satisfy the assumptions of the previous theorem, then <sup id="cite_ref-FOOTNOTEKlartag2008Theorem_1_35-0" class="reference"><a href="#cite_note-FOOTNOTEKlartag2008Theorem_1-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\mathbb {P} \left(a\leq {\frac {X_{1}+\cdots +X_{n}}{\sqrt {n}}}\leq b\right)-{\frac {1}{\sqrt {2\pi }}}\int _{a}^{b}e^{-{\frac {1}{2}}t^{2}}\,dt\right|\leq {\frac {C}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <msqrt> <mi>n</mi> </msqrt> </mfrac> </mrow> <mo>≤</mo> <mi>b</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\mathbb {P} \left(a\leq {\frac {X_{1}+\cdots +X_{n}}{\sqrt {n}}}\leq b\right)-{\frac {1}{\sqrt {2\pi }}}\int _{a}^{b}e^{-{\frac {1}{2}}t^{2}}\,dt\right|\leq {\frac {C}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c68c5077344e1cc79881c190943e766a8f542e00" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:56.257ex; height:6.843ex;" alt="{\displaystyle \left|\mathbb {P} \left(a\leq {\frac {X_{1}+\cdots +X_{n}}{\sqrt {n}}}\leq b\right)-{\frac {1}{\sqrt {2\pi }}}\int _{a}^{b}e^{-{\frac {1}{2}}t^{2}}\,dt\right|\leq {\frac {C}{n}}}"></span> for all <span class="texhtml"><i>a</i> < <i>b</i></span>; here <span class="texhtml mvar" style="font-style:italic;">C</span> is a <a href="/wiki/Mathematical_constant" title="Mathematical constant">universal (absolute) constant</a>. Moreover, for every <span class="texhtml"><i>c</i><sub>1</sub>, ..., <i>c<sub>n</sub></i> ∈ <b>R</b></span> such that <span class="texhtml"><i>c</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span> + ⋯ + <i>c</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>n</i></sub></span></span> = 1</span>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\mathbb {P} \left(a\leq c_{1}X_{1}+\cdots +c_{n}X_{n}\leq b\right)-{\frac {1}{\sqrt {2\pi }}}\int _{a}^{b}e^{-{\frac {1}{2}}t^{2}}\,dt\right|\leq C\left(c_{1}^{4}+\dots +c_{n}^{4}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>≤</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <mi>b</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mrow> <mo>|</mo> </mrow> <mo>≤</mo> <mi>C</mi> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\mathbb {P} \left(a\leq c_{1}X_{1}+\cdots +c_{n}X_{n}\leq b\right)-{\frac {1}{\sqrt {2\pi }}}\int _{a}^{b}e^{-{\frac {1}{2}}t^{2}}\,dt\right|\leq C\left(c_{1}^{4}+\dots +c_{n}^{4}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbc1a71d4ee2ee74bfd8b7b67959d79c171afc08" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:73.501ex; height:6.843ex;" alt="{\displaystyle \left|\mathbb {P} \left(a\leq c_{1}X_{1}+\cdots +c_{n}X_{n}\leq b\right)-{\frac {1}{\sqrt {2\pi }}}\int _{a}^{b}e^{-{\frac {1}{2}}t^{2}}\,dt\right|\leq C\left(c_{1}^{4}+\dots +c_{n}^{4}\right).}"></span> </p> </div> <p>The distribution of <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>X</i><sub>1</sub> + ⋯ + <i>X<sub>n</sub></i></span><span class="sr-only">/</span><span class="den"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>n</i></span></span></span></span>⁠</span></span> need not be approximately normal (in fact, it can be uniform).<sup id="cite_ref-FOOTNOTEKlartag2007Theorem_1.1_36-0" class="reference"><a href="#cite_note-FOOTNOTEKlartag2007Theorem_1.1-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> However, the distribution of <span class="texhtml"><i>c</i><sub>1</sub><i>X</i><sub>1</sub> + ⋯ + <i>c<sub>n</sub>X<sub>n</sub></i></span> is close to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {N}}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathcal {N}}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba37ff02211fb81c813065b489b78ec9df951e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:7.505ex; height:3.009ex;" alt="{\textstyle {\mathcal {N}}(0,1)}"></span> (in the total variation distance) for most vectors <span class="texhtml">(<i>c</i><sub>1</sub>, ..., <i>c<sub>n</sub></i>)</span> according to the uniform distribution on the sphere <span class="texhtml"><i>c</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span> + ⋯ + <i>c</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>n</i></sub></span></span> = 1</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Lacunary_trigonometric_series">Lacunary trigonometric series</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=23" title="Edit section: Lacunary trigonometric series"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem (<a href="/wiki/Rapha%C3%ABl_Salem" title="Raphaël Salem">Salem</a>–<a href="/wiki/Antoni_Zygmund" title="Antoni Zygmund">Zygmund</a>)</strong><span class="theoreme-tiret"> — </span>Let <span class="texhtml mvar" style="font-style:italic;">U</span> be a random variable distributed uniformly on <span class="texhtml">(0,2π)</span>, and <span class="texhtml"><i>X<sub>k</sub></i> = <i>r<sub>k</sub></i> cos(<i>n<sub>k</sub>U</i> + <i>a<sub>k</sub></i>)</span>, where </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">n<sub>k</sub></span> satisfy the lacunarity condition: there exists <span class="texhtml"><i>q</i> > 1</span> such that <span class="texhtml"><i>n</i><sub><i>k</i> + 1</sub> ≥ <i>qn</i><sub><i>k</i></sub></span> for all <span class="texhtml mvar" style="font-style:italic;">k</span>,</li> <li><span class="texhtml mvar" style="font-style:italic;">r<sub>k</sub></span> are such that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}^{2}+r_{2}^{2}+\cdots =\infty \quad {\text{ and }}\quad {\frac {r_{k}^{2}}{r_{1}^{2}+\cdots +r_{k}^{2}}}\to 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mi mathvariant="normal">∞</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo stretchy="false">→</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}^{2}+r_{2}^{2}+\cdots =\infty \quad {\text{ and }}\quad {\frac {r_{k}^{2}}{r_{1}^{2}+\cdots +r_{k}^{2}}}\to 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d87209928a7946d7d202155512b294c79e72ada" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.49ex; height:7.176ex;" alt="{\displaystyle r_{1}^{2}+r_{2}^{2}+\cdots =\infty \quad {\text{ and }}\quad {\frac {r_{k}^{2}}{r_{1}^{2}+\cdots +r_{k}^{2}}}\to 0,}"></span></li> <li><span class="texhtml">0 ≤ <i>a</i><sub><i>k</i></sub> < 2π</span>.</li></ul> <p>Then<sup id="cite_ref-Zygmund_37-0" class="reference"><a href="#cite_note-Zygmund-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEGaposhkin1966Theorem_2.1.13_38-0" class="reference"><a href="#cite_note-FOOTNOTEGaposhkin1966Theorem_2.1.13-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {X_{1}+\cdots +X_{k}}{\sqrt {r_{1}^{2}+\cdots +r_{k}^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <msqrt> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {X_{1}+\cdots +X_{k}}{\sqrt {r_{1}^{2}+\cdots +r_{k}^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f65a0da4b413fce55317ae98f18905fe91442a2c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:15.804ex; height:8.009ex;" alt="{\displaystyle {\frac {X_{1}+\cdots +X_{k}}{\sqrt {r_{1}^{2}+\cdots +r_{k}^{2}}}}}"></span> converges in distribution to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {N}}{\big (}0,{\frac {1}{2}}{\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathcal {N}}{\big (}0,{\frac {1}{2}}{\big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8c8d6598f79fac7b8c5f683f4319e95306e0ce9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-left: -0.062ex; width:8.321ex; height:3.509ex;" alt="{\textstyle {\mathcal {N}}{\big (}0,{\frac {1}{2}}{\big )}}"></span>. </p> </div> <div class="mw-heading mw-heading3"><h3 id="Gaussian_polytopes">Gaussian polytopes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=24" title="Edit section: Gaussian polytopes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem</strong><span class="theoreme-tiret"> — </span>Let <span class="texhtml"><i>A</i><sub>1</sub>, ..., <i>A</i><sub><i>n</i></sub></span> be independent random points on the plane <span class="texhtml"><b>R</b><sup>2</sup></span> each having the two-dimensional standard normal distribution. Let <span class="texhtml mvar" style="font-style:italic;">K<sub>n</sub></span> be the <a href="/wiki/Convex_hull" title="Convex hull">convex hull</a> of these points, and <span class="texhtml mvar" style="font-style:italic;">X<sub>n</sub></span> the area of <span class="texhtml mvar" style="font-style:italic;">K<sub>n</sub></span> Then<sup id="cite_ref-FOOTNOTEBárányVu2007Theorem_1.1_39-0" class="reference"><a href="#cite_note-FOOTNOTEBárányVu2007Theorem_1.1-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {X_{n}-\operatorname {E} (X_{n})}{\sqrt {\operatorname {Var} (X_{n})}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <msqrt> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {X_{n}-\operatorname {E} (X_{n})}{\sqrt {\operatorname {Var} (X_{n})}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98854722b040253e6fe84713c8d08c0307afb0a7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:13.354ex; height:7.009ex;" alt="{\displaystyle {\frac {X_{n}-\operatorname {E} (X_{n})}{\sqrt {\operatorname {Var} (X_{n})}}}}"></span> converges in distribution to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {N}}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathcal {N}}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba37ff02211fb81c813065b489b78ec9df951e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:7.505ex; height:3.009ex;" alt="{\textstyle {\mathcal {N}}(0,1)}"></span> as <span class="texhtml mvar" style="font-style:italic;">n</span> tends to infinity. </p> </div> <p>The same also holds in all dimensions greater than 2. </p><p>The <a href="/wiki/Convex_polytope" title="Convex polytope">polytope</a> <span class="texhtml mvar" style="font-style:italic;">K<sub>n</sub></span> is called a Gaussian random polytope. </p><p>A similar result holds for the number of vertices (of the Gaussian polytope), the number of edges, and in fact, faces of all dimensions.<sup id="cite_ref-FOOTNOTEBárányVu2007Theorem_1.2_40-0" class="reference"><a href="#cite_note-FOOTNOTEBárányVu2007Theorem_1.2-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Linear_functions_of_orthogonal_matrices">Linear functions of orthogonal matrices</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=25" title="Edit section: Linear functions of orthogonal matrices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A linear function of a matrix <span class="texhtml"><b>M</b></span> is a linear combination of its elements (with given coefficients), <span class="texhtml"><b>M</b> ↦ tr(<b>AM</b>)</span> where <span class="texhtml"><b>A</b></span> is the matrix of the coefficients; see <a href="/wiki/Trace_(linear_algebra)#Inner_product" title="Trace (linear algebra)">Trace (linear algebra)#Inner product</a>. </p><p>A random <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal matrix</a> is said to be distributed uniformly, if its distribution is the normalized <a href="/wiki/Haar_measure" title="Haar measure">Haar measure</a> on the <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a> <span class="texhtml">O(<i>n</i>,<b>R</b>)</span>; see <a href="/wiki/Rotation_matrix#Uniform_random_rotation_matrices" title="Rotation matrix">Rotation matrix#Uniform random rotation matrices</a>. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem</strong><span class="theoreme-tiret"> — </span>Let <span class="texhtml"><b>M</b></span> be a random orthogonal <span class="texhtml"><i>n</i> × <i>n</i></span> matrix distributed uniformly, and <span class="texhtml"><b>A</b></span> a fixed <span class="texhtml"><i>n</i> × <i>n</i></span> matrix such that <span class="texhtml">tr(<b>AA</b>*) = <i>n</i></span>, and let <span class="texhtml"><i>X</i> = tr(<b>AM</b>)</span>. Then<sup id="cite_ref-Meckes_41-0" class="reference"><a href="#cite_note-Meckes-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> the distribution of <span class="texhtml mvar" style="font-style:italic;">X</span> is close to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {N}}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathcal {N}}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba37ff02211fb81c813065b489b78ec9df951e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:7.505ex; height:3.009ex;" alt="{\textstyle {\mathcal {N}}(0,1)}"></span> in the total variation metric up to<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="what does up to mean here (June 2012)">clarification needed</span></a></i>]</sup> <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">2<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">3</span></span></span><span class="sr-only">/</span><span class="den"><i>n</i> − 1</span></span>⁠</span></span>. </p> </div> <div class="mw-heading mw-heading3"><h3 id="Subsequences">Subsequences</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=26" title="Edit section: Subsequences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem</strong><span class="theoreme-tiret"> — </span>Let random variables <span class="texhtml"><i>X</i><sub>1</sub>, <i>X</i><sub>2</sub>, ... ∈ <i>L</i><sub>2</sub>(Ω)</span> be such that <span class="texhtml"><i>X<sub>n</sub></i> → 0</span> <a href="/wiki/Weak_convergence_(Hilbert_space)" title="Weak convergence (Hilbert space)">weakly</a> in <span class="texhtml"><i>L</i><sub>2</sub>(Ω)</span> and <span class="texhtml"><i>X</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>n</i></sub></span></span> → 1</span> weakly in <span class="texhtml"><i>L</i><sub>1</sub>(Ω)</span>. Then there exist integers <span class="texhtml"><i>n</i><sub>1</sub> < <i>n</i><sub>2</sub> < ⋯</span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {X_{n_{1}}+\cdots +X_{n_{k}}}{\sqrt {k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> </mrow> <msqrt> <mi>k</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {X_{n_{1}}+\cdots +X_{n_{k}}}{\sqrt {k}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2926d591f2008d629880d119762ac66b916f18f2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:17.217ex; height:6.676ex;" alt="{\displaystyle {\frac {X_{n_{1}}+\cdots +X_{n_{k}}}{\sqrt {k}}}}"></span> converges in distribution to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {N}}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathcal {N}}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba37ff02211fb81c813065b489b78ec9df951e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:7.505ex; height:3.009ex;" alt="{\textstyle {\mathcal {N}}(0,1)}"></span> as <span class="texhtml mvar" style="font-style:italic;">k</span> tends to infinity.<sup id="cite_ref-FOOTNOTEGaposhkin1966Sect._1.5_42-0" class="reference"><a href="#cite_note-FOOTNOTEGaposhkin1966Sect._1.5-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> </p> </div> <div class="mw-heading mw-heading3"><h3 id="Random_walk_on_a_crystal_lattice">Random walk on a crystal lattice</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=27" title="Edit section: Random walk on a crystal lattice"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The central limit theorem may be established for the simple <a href="/wiki/Random_walk" title="Random walk">random walk</a> on a crystal lattice (an infinite-fold abelian covering graph over a finite graph), and is used for design of crystal structures.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications_and_examples">Applications and examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=28" title="Edit section: Applications and examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A simple example of the central limit theorem is rolling many identical, unbiased dice. The distribution of the sum (or average) of the rolled numbers will be well approximated by a normal distribution. Since real-world quantities are often the balanced sum of many unobserved random events, the central limit theorem also provides a partial explanation for the prevalence of the normal probability distribution. It also justifies the approximation of large-sample <a href="/wiki/Statistic" title="Statistic">statistics</a> to the normal distribution in controlled experiments. </p> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tnone center"><div class="thumbinner multiimageinner" style="width:822px;max-width:822px"><div class="trow"><div class="tsingle" style="width:329px;max-width:329px"><div class="thumbimage" style="height:391px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Dice_sum_central_limit_theorem.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Dice_sum_central_limit_theorem.svg/327px-Dice_sum_central_limit_theorem.svg.png" decoding="async" width="327" height="392" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Dice_sum_central_limit_theorem.svg/491px-Dice_sum_central_limit_theorem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Dice_sum_central_limit_theorem.svg/654px-Dice_sum_central_limit_theorem.svg.png 2x" data-file-width="512" data-file-height="614" /></a></span></div><div class="thumbcaption">Comparison of probability density functions <span class="texhtml"><i>p</i>(<i>k</i>)</span> for the sum of <span class="texhtml mvar" style="font-style:italic;">n</span> fair 6-sided dice to show their convergence to a normal distribution with increasing <span class="texhtml mvar" style="font-style:italic;">n</span>, in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).</div></div><div class="tsingle" style="width:489px;max-width:489px"><div class="thumbimage" style="height:391px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Empirical_CLT_-_Figure_-_040711.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Empirical_CLT_-_Figure_-_040711.jpg/487px-Empirical_CLT_-_Figure_-_040711.jpg" decoding="async" width="487" height="392" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/2/2d/Empirical_CLT_-_Figure_-_040711.jpg 1.5x" data-file-width="669" data-file-height="538" /></a></span></div><div class="thumbcaption">This figure demonstrates the central limit theorem. The sample means are generated using a random number generator, which draws numbers between 0 and 100 from a uniform probability distribution. It illustrates that increasing sample sizes result in the 500 measured sample means being more closely distributed about the population mean (50 in this case). It also compares the observed distributions with the distributions that would be expected for a normalized Gaussian distribution, and shows the <a href="/wiki/Pearson%27s_chi-squared_test" title="Pearson's chi-squared test">chi-squared</a> values that quantify the goodness of the fit (the fit is good if the reduced <a href="/wiki/Pearson%27s_chi-squared_test" title="Pearson's chi-squared test">chi-squared</a> value is less than or approximately equal to one). The input into the normalized Gaussian function is the mean of sample means (~50) and the mean sample standard deviation divided by the square root of the sample size (~28.87/<span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>n</i></span></span></span>), which is called the standard deviation of the mean (since it refers to the spread of sample means).</div></div></div></div></div> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Mean-of-the-outcomes-of-rolling-a-fair-coin-n-times.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Mean-of-the-outcomes-of-rolling-a-fair-coin-n-times.svg/820px-Mean-of-the-outcomes-of-rolling-a-fair-coin-n-times.svg.png" decoding="async" width="820" height="818" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Mean-of-the-outcomes-of-rolling-a-fair-coin-n-times.svg/1230px-Mean-of-the-outcomes-of-rolling-a-fair-coin-n-times.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/75/Mean-of-the-outcomes-of-rolling-a-fair-coin-n-times.svg/1640px-Mean-of-the-outcomes-of-rolling-a-fair-coin-n-times.svg.png 2x" data-file-width="472" data-file-height="471" /></a><figcaption>Another simulation using the binomial distribution. Random 0s and 1s were generated, and then their means calculated for sample sizes ranging from 1 to 2048. Note that as the sample size increases the tails become thinner and the distribution becomes more concentrated around the mean.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Regression">Regression</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=29" title="Edit section: Regression"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a>, and in particular <a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">ordinary least squares</a>, specifies that a <a href="/wiki/Dependent_variable" class="mw-redirect" title="Dependent variable">dependent variable</a> depends according to some function upon one or more <a href="/wiki/Independent_variable" class="mw-redirect" title="Independent variable">independent variables</a>, with an additive <a href="/wiki/Errors_and_residuals_in_statistics" class="mw-redirect" title="Errors and residuals in statistics">error term</a>. Various types of statistical inference on the regression assume that the error term is normally distributed. This assumption can be justified by assuming that the error term is actually the sum of many independent error terms; even if the individual error terms are not normally distributed, by the central limit theorem their sum can be well approximated by a normal distribution. </p> <div class="mw-heading mw-heading3"><h3 id="Other_illustrations">Other illustrations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=30" title="Edit section: Other illustrations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Illustration_of_the_central_limit_theorem" title="Illustration of the central limit theorem">Illustration of the central limit theorem</a></div> <p>Given its importance to statistics, a number of papers and computer packages are available that demonstrate the convergence involved in the central limit theorem.<sup id="cite_ref-Marasinghe_45-0" class="reference"><a href="#cite_note-Marasinghe-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=31" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dutch mathematician <a href="/wiki/Henk_Tijms" title="Henk Tijms">Henk Tijms</a> writes:<sup id="cite_ref-Tijms_46-0" class="reference"><a href="#cite_note-Tijms-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>The central limit theorem has an interesting history. The first version of this theorem was postulated by the French-born mathematician <a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a> who, in a remarkable article published in 1733, used the normal distribution to approximate the distribution of the number of heads resulting from many tosses of a fair coin. This finding was far ahead of its time, and was nearly forgotten until the famous French mathematician <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> rescued it from obscurity in his monumental work <i>Théorie analytique des probabilités</i>, which was published in 1812. Laplace expanded De Moivre's finding by approximating the binomial distribution with the normal distribution. But as with De Moivre, Laplace's finding received little attention in his own time. It was not until the nineteenth century was at an end that the importance of the central limit theorem was discerned, when, in 1901, Russian mathematician <a href="/wiki/Aleksandr_Lyapunov" title="Aleksandr Lyapunov">Aleksandr Lyapunov</a> defined it in general terms and proved precisely how it worked mathematically. Nowadays, the central limit theorem is considered to be the unofficial sovereign of probability theory.</p></blockquote> <p>Sir <a href="/wiki/Francis_Galton" title="Francis Galton">Francis Galton</a> described the Central Limit Theorem in this way:<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the "Law of Frequency of Error". The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.</p></blockquote> <p>The actual term "central limit theorem" (in German: "zentraler Grenzwertsatz") was first used by <a href="/wiki/George_P%C3%B3lya" title="George Pólya">George Pólya</a> in 1920 in the title of a paper.<sup id="cite_ref-Polya1920_48-0" class="reference"><a href="#cite_note-Polya1920-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-LC1986_49-0" class="reference"><a href="#cite_note-LC1986-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> Pólya referred to the theorem as "central" due to its importance in probability theory. According to Le Cam, the French school of probability interprets the word <i>central</i> in the sense that "it describes the behaviour of the centre of the distribution as opposed to its tails".<sup id="cite_ref-LC1986_49-1" class="reference"><a href="#cite_note-LC1986-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> The abstract of the paper <i>On the central limit theorem of calculus of probability and the problem of moments</i> by Pólya<sup id="cite_ref-Polya1920_48-1" class="reference"><a href="#cite_note-Polya1920-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> in 1920 translates as follows. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>The occurrence of the Gaussian probability density <span class="texhtml">1 = <i>e</i><sup>−<i>x</i><sup>2</sup></sup></span> in repeated experiments, in errors of measurements, which result in the combination of very many and very small elementary errors, in diffusion processes etc., can be explained, as is well-known, by the very same limit theorem, which plays a central role in the calculus of probability. The actual discoverer of this limit theorem is to be named Laplace; it is likely that its rigorous proof was first given by Tschebyscheff and its sharpest formulation can be found, as far as I am aware of, in an article by <a href="/wiki/Aleksandr_Lyapunov" title="Aleksandr Lyapunov">Liapounoff</a>. ...</p></blockquote> <p>A thorough account of the theorem's history, detailing Laplace's foundational work, as well as <a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a>'s, <a href="/wiki/Friedrich_Bessel" class="mw-redirect" title="Friedrich Bessel">Bessel</a>'s and <a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a>'s contributions, is provided by Hald.<sup id="cite_ref-Hald_50-0" class="reference"><a href="#cite_note-Hald-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> Two historical accounts, one covering the development from Laplace to Cauchy, the second the contributions by <a href="/wiki/Richard_von_Mises" title="Richard von Mises">von Mises</a>, <a href="/wiki/George_P%C3%B3lya" title="George Pólya">Pólya</a>, <a href="/wiki/Jarl_Waldemar_Lindeberg" title="Jarl Waldemar Lindeberg">Lindeberg</a>, <a href="/wiki/Paul_L%C3%A9vy_(mathematician)" title="Paul Lévy (mathematician)">Lévy</a>, and <a href="/wiki/Harald_Cram%C3%A9r" title="Harald Cramér">Cramér</a> during the 1920s, are given by Hans Fischer.<sup id="cite_ref-FOOTNOTEFischer2011Chapter_2;_Chapter_5.2_51-0" class="reference"><a href="#cite_note-FOOTNOTEFischer2011Chapter_2;_Chapter_5.2-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> Le Cam describes a period around 1935.<sup id="cite_ref-LC1986_49-2" class="reference"><a href="#cite_note-LC1986-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> Bernstein<sup id="cite_ref-Bernstein_52-0" class="reference"><a href="#cite_note-Bernstein-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> presents a historical discussion focusing on the work of <a href="/wiki/Pafnuty_Chebyshev" title="Pafnuty Chebyshev">Pafnuty Chebyshev</a> and his students <a href="/wiki/Andrey_Markov" title="Andrey Markov">Andrey Markov</a> and <a href="/wiki/Aleksandr_Lyapunov" title="Aleksandr Lyapunov">Aleksandr Lyapunov</a> that led to the first proofs of the CLT in a general setting. </p><p>A curious footnote to the history of the Central Limit Theorem is that a proof of a result similar to the 1922 Lindeberg CLT was the subject of <a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a>'s 1934 Fellowship Dissertation for <a href="/wiki/King%27s_College,_Cambridge" title="King's College, Cambridge">King's College</a> at the <a href="/wiki/University_of_Cambridge" title="University of Cambridge">University of Cambridge</a>. Only after submitting the work did Turing learn it had already been proved. Consequently, Turing's dissertation was not published.<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=32" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Asymptotic_equipartition_property" title="Asymptotic equipartition property">Asymptotic equipartition property</a></li> <li><a href="/wiki/Asymptotic_distribution" title="Asymptotic distribution">Asymptotic distribution</a></li> <li><a href="/wiki/Bates_distribution" title="Bates distribution">Bates distribution</a></li> <li><a href="/wiki/Benford%27s_law" title="Benford's law">Benford's law</a> – result of extension of CLT to product of random variables.</li> <li><a href="/wiki/Berry%E2%80%93Esseen_theorem" title="Berry–Esseen theorem">Berry–Esseen theorem</a></li> <li><a href="/wiki/Central_limit_theorem_for_directional_statistics" title="Central limit theorem for directional statistics">Central limit theorem for directional statistics</a> – Central limit theorem applied to the case of directional statistics</li> <li><a href="/wiki/Delta_method" title="Delta method">Delta method</a> – to compute the limit distribution of a function of a random variable.</li> <li><a href="/wiki/Erd%C5%91s%E2%80%93Kac_theorem" title="Erdős–Kac theorem">Erdős–Kac theorem</a> – connects the number of prime factors of an integer with the normal probability distribution</li> <li><a href="/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem" title="Fisher–Tippett–Gnedenko theorem">Fisher–Tippett–Gnedenko theorem</a> – limit theorem for extremum values (such as <span class="texhtml">max{<i>X<sub>n</sub></i>}</span>)</li> <li><a href="/wiki/Irwin%E2%80%93Hall_distribution" title="Irwin–Hall distribution">Irwin–Hall distribution</a></li> <li><a href="/wiki/Markov_chain_central_limit_theorem" title="Markov chain central limit theorem">Markov chain central limit theorem</a></li> <li><a href="/wiki/Normal_distribution" title="Normal distribution">Normal distribution</a></li> <li><a href="/wiki/Tweedie_distribution" title="Tweedie distribution">Tweedie convergence theorem</a> – a theorem that can be considered to bridge between the central limit theorem and the <a href="/wiki/Poisson_convergence_theorem" class="mw-redirect" title="Poisson convergence theorem">Poisson convergence theorem</a><sup id="cite_ref-Jørgensen-1997_54-0" class="reference"><a href="#cite_note-Jørgensen-1997-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Donsker%27s_theorem" title="Donsker's theorem">Donsker's theorem</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=33" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-FOOTNOTEFischer2011[[Category:Wikipedia_articles_needing_page_number_citations_from_July_2023]]<sup_class="noprint_Inline-Template_"_style="white-space:nowrap;">&#91;<i>[[Wikipedia:Citing_sources|<span_title="This_citation_requires_a_reference_to_the_specific_page_or_range_of_pages_in_which_the_material_appears.&#32;(July_2023)">page&nbsp;needed</span>]]</i>&#93;</sup>-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFischer2011[[Category:Wikipedia_articles_needing_page_number_citations_from_July_2023]]<sup_class="noprint_Inline-Template_"_style="white-space:nowrap;">&#91;<i>[[Wikipedia:Citing_sources|<span_title="This_citation_requires_a_reference_to_the_specific_page_or_range_of_pages_in_which_the_material_appears.&#32;(July_2023)">page&nbsp;needed</span>]]</i>&#93;</sup>_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFischer2011">Fischer (2011)</a>, p. <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (July 2023)">page needed</span></a></i>]</sup>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMontgomeryRunger2014" class="citation book cs1">Montgomery, Douglas C.; Runger, George C. (2014). <i>Applied Statistics and Probability for Engineers</i> (6th ed.). Wiley. p. 241. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781118539712" title="Special:BookSources/9781118539712"><bdi>9781118539712</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Statistics+and+Probability+for+Engineers&rft.pages=241&rft.edition=6th&rft.pub=Wiley&rft.date=2014&rft.isbn=9781118539712&rft.aulast=Montgomery&rft.aufirst=Douglas+C.&rft.au=Runger%2C+George+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRouaud2013" class="citation book cs1">Rouaud, Mathieu (2013). <a rel="nofollow" class="external text" href="http://www.incertitudes.fr/book.pdf"><i>Probability, Statistics and Estimation</i></a> <span class="cs1-format">(PDF)</span>. p. 10. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://www.incertitudes.fr/book.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-10-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability%2C+Statistics+and+Estimation&rft.pages=10&rft.date=2013&rft.aulast=Rouaud&rft.aufirst=Mathieu&rft_id=http%3A%2F%2Fwww.incertitudes.fr%2Fbook.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEBillingsley1995357-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBillingsley1995357_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBillingsley1995">Billingsley (1995)</a>, p. 357.</span> </li> <li id="cite_note-FOOTNOTEBauer2001199Theorem_30.13-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBauer2001199Theorem_30.13_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBauer2001">Bauer (2001)</a>, p. 199, Theorem 30.13.</span> </li> <li id="cite_note-FOOTNOTEBillingsley1995362-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBillingsley1995362_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBillingsley1995">Billingsley (1995)</a>, p. 362.</span> </li> <li id="cite_note-vanderVaart-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-vanderVaart_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-vanderVaart_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_der_Vaart1998" class="citation book cs1">van der Vaart, A.W. (1998). <i>Asymptotic statistics</i>. New York, NY: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-49603-2" title="Special:BookSources/978-0-521-49603-2"><bdi>978-0-521-49603-2</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/98015176">98015176</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Asymptotic+statistics&rft.place=New+York%2C+NY&rft.pub=Cambridge+University+Press&rft.date=1998&rft_id=info%3Alccn%2F98015176&rft.isbn=978-0-521-49603-2&rft.aulast=van+der+Vaart&rft.aufirst=A.W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO’Donnell2014" class="citation web cs1"><a href="/wiki/Ryan_O%27Donnell_(computer_scientist)" title="Ryan O'Donnell (computer scientist)">O’Donnell, Ryan</a> (2014). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190408054104/http://www.contrib.andrew.cmu.edu/~ryanod/?p=866">"Theorem 5.38"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.contrib.andrew.cmu.edu/~ryanod/?p=866">the original</a> on 2019-04-08<span class="reference-accessdate">. Retrieved <span class="nowrap">2017-10-18</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Theorem+5.38&rft.date=2014&rft.aulast=O%E2%80%99Donnell&rft.aufirst=Ryan&rft_id=http%3A%2F%2Fwww.contrib.andrew.cmu.edu%2F~ryanod%2F%3Fp%3D866&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBentkus2005" class="citation journal cs1">Bentkus, V. (2005). "A Lyapunov-type bound in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a713426956296f1668fce772df3c60b9dde8a685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{d}}"></span>". <i>Theory Probab. Appl</i>. <b>49</b> (2): 311–323. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2FS0040585X97981123">10.1137/S0040585X97981123</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theory+Probab.+Appl.&rft.atitle=A+Lyapunov-type+bound+in+MATH+RENDER+ERROR&rft.volume=49&rft.issue=2&rft.pages=311-323&rft.date=2005&rft_id=info%3Adoi%2F10.1137%2FS0040585X97981123&rft.aulast=Bentkus&rft.aufirst=V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLe_Cam1986" class="citation journal cs1">Le Cam, L. (February 1986). "The Central Limit Theorem around 1935". <i>Statistical Science</i>. <b>1</b> (1): 78–91. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2245503">2245503</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistical+Science&rft.atitle=The+Central+Limit+Theorem+around+1935&rft.volume=1&rft.issue=1&rft.pages=78-91&rft.date=1986-02&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2245503%23id-name%3DJSTOR&rft.aulast=Le+Cam&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLévy1937" class="citation book cs1">Lévy, Paul (1937). <i>Theorie de l'addition des variables aleatoires [Combination theory of unpredictable variables]</i>. Paris: Gauthier-Villars.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theorie+de+l%27addition+des+variables+aleatoires+%5BCombination+theory+of+unpredictable+variables%5D&rft.place=Paris&rft.pub=Gauthier-Villars&rft.date=1937&rft.aulast=L%C3%A9vy&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGnedenkoKologorovDoobHsu1968" class="citation book cs1">Gnedenko, Boris Vladimirovich; Kologorov, Andreĭ Nikolaevich; Doob, Joseph L.; Hsu, Pao-Lu (1968). <i>Limit distributions for sums of independent random variables</i>. Reading, MA: Addison-wesley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Limit+distributions+for+sums+of+independent+random+variables&rft.place=Reading%2C+MA&rft.pub=Addison-wesley&rft.date=1968&rft.aulast=Gnedenko&rft.aufirst=Boris+Vladimirovich&rft.au=Kologorov%2C+Andre%C4%AD+Nikolaevich&rft.au=Doob%2C+Joseph+L.&rft.au=Hsu%2C+Pao-Lu&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNolan2020" class="citation book cs1">Nolan, John P. (2020). <a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-030-52915-4"><i>Univariate stable distributions, Models for Heavy Tailed Data</i></a>. Springer Series in Operations Research and Financial Engineering. Switzerland: Springer. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-030-52915-4">10.1007/978-3-030-52915-4</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-52914-7" title="Special:BookSources/978-3-030-52914-7"><bdi>978-3-030-52914-7</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:226648987">226648987</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Univariate+stable+distributions%2C+Models+for+Heavy+Tailed+Data&rft.place=Switzerland&rft.series=Springer+Series+in+Operations+Research+and+Financial+Engineering&rft.pub=Springer&rft.date=2020&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A226648987%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-3-030-52915-4&rft.isbn=978-3-030-52914-7&rft.aulast=Nolan&rft.aufirst=John+P.&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%2F978-3-030-52915-4&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEBillingsley1995Theorem_27.4-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBillingsley1995Theorem_27.4_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBillingsley1995">Billingsley (1995)</a>, Theorem 27.4.</span> </li> <li id="cite_note-FOOTNOTEDurrett2004Sect._7.7(c),_Theorem_7.8-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDurrett2004Sect._7.7(c),_Theorem_7.8_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDurrett2004">Durrett (2004)</a>, Sect. 7.7(c), Theorem 7.8.</span> </li> <li id="cite_note-FOOTNOTEDurrett2004Sect._7.7,_Theorem_7.4-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDurrett2004Sect._7.7,_Theorem_7.4_16-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDurrett2004">Durrett (2004)</a>, Sect. 7.7, Theorem 7.4.</span> </li> <li id="cite_note-FOOTNOTEBillingsley1995Theorem_35.12-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBillingsley1995Theorem_35.12_17-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBillingsley1995">Billingsley (1995)</a>, Theorem 35.12.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLemons2003" class="citation book cs1">Lemons, Don (2003). <a rel="nofollow" class="external text" href="https://jhupbooks.press.jhu.edu/content/introduction-stochastic-processes-physics"><i>An Introduction to Stochastic Processes in Physics</i></a>. Johns Hopkins University Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.56021%2F9780801868665">10.56021/9780801868665</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780801876387" title="Special:BookSources/9780801876387"><bdi>9780801876387</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2016-08-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Stochastic+Processes+in+Physics&rft.pub=Johns+Hopkins+University+Press&rft.date=2003&rft_id=info%3Adoi%2F10.56021%2F9780801868665&rft.isbn=9780801876387&rft.aulast=Lemons&rft.aufirst=Don&rft_id=https%3A%2F%2Fjhupbooks.press.jhu.edu%2Fcontent%2Fintroduction-stochastic-processes-physics&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-stein1972-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-stein1972_19-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStein1972" class="citation journal cs1"><a href="/wiki/Charles_Stein_(statistician)" class="mw-redirect" title="Charles Stein (statistician)">Stein, C.</a> (1972). <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.bsmsp/1200514239">"A bound for the error in the normal approximation to the distribution of a sum of dependent random variables"</a>. <i>Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability</i>. <b>6</b> (2): 583–602. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0402873">0402873</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0278.60026">0278.60026</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Sixth+Berkeley+Symposium+on+Mathematical+Statistics+and+Probability&rft.atitle=A+bound+for+the+error+in+the+normal+approximation+to+the+distribution+of+a+sum+of+dependent+random+variables&rft.volume=6&rft.issue=2&rft.pages=583-602&rft.date=1972&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0278.60026%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D402873%23id-name%3DMR&rft.aulast=Stein&rft.aufirst=C.&rft_id=http%3A%2F%2Fprojecteuclid.org%2Feuclid.bsmsp%2F1200514239&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChenGoldsteinShao2011" class="citation book cs1">Chen, L. H. Y.; Goldstein, L.; Shao, Q. M. (2011). <i>Normal approximation by Stein's method</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-15006-7" title="Special:BookSources/978-3-642-15006-7"><bdi>978-3-642-15006-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Normal+approximation+by+Stein%27s+method&rft.pub=Springer&rft.date=2011&rft.isbn=978-3-642-15006-7&rft.aulast=Chen&rft.aufirst=L.+H.+Y.&rft.au=Goldstein%2C+L.&rft.au=Shao%2C+Q.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-ABBN-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-ABBN_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArtsteinBallBartheNaor2004" class="citation journal cs1"><a href="/wiki/Shiri_Artstein" title="Shiri Artstein">Artstein, S.</a>; <a href="/wiki/Keith_Martin_Ball" title="Keith Martin Ball">Ball, K.</a>; <a href="/wiki/Franck_Barthe" title="Franck Barthe">Barthe, F.</a>; <a href="/wiki/Assaf_Naor" title="Assaf Naor">Naor, A.</a> (2004). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0894-0347-04-00459-X">"Solution of Shannon's Problem on the Monotonicity of Entropy"</a>. <i>Journal of the American Mathematical Society</i>. <b>17</b> (4): 975–982. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0894-0347-04-00459-X">10.1090/S0894-0347-04-00459-X</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+American+Mathematical+Society&rft.atitle=Solution+of+Shannon%27s+Problem+on+the+Monotonicity+of+Entropy&rft.volume=17&rft.issue=4&rft.pages=975-982&rft.date=2004&rft_id=info%3Adoi%2F10.1090%2FS0894-0347-04-00459-X&rft.aulast=Artstein&rft.aufirst=S.&rft.au=Ball%2C+K.&rft.au=Barthe%2C+F.&rft.au=Naor%2C+A.&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252FS0894-0347-04-00459-X&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrewer1985" class="citation journal cs1">Brewer, J.K. (1985). "Behavioral statistics textbooks: Source of myths and misconceptions?". <i>Journal of Educational Statistics</i>. <b>10</b> (3): 252–268. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3102%2F10769986010003252">10.3102/10769986010003252</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119611584">119611584</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Educational+Statistics&rft.atitle=Behavioral+statistics+textbooks%3A+Source+of+myths+and+misconceptions%3F&rft.volume=10&rft.issue=3&rft.pages=252-268&rft.date=1985&rft_id=info%3Adoi%2F10.3102%2F10769986010003252&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119611584%23id-name%3DS2CID&rft.aulast=Brewer&rft.aufirst=J.K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">Yu, C.; Behrens, J.; Spencer, A. Identification of Misconception in the Central Limit Theorem and Related Concepts, <i>American Educational Research Association</i> lecture 19 April 1995</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSotosVanhoofVan_den_NoortgateOnghena2007" class="citation journal cs1">Sotos, A.E.C.; Vanhoof, S.; Van den Noortgate, W.; Onghena, P. (2007). <a rel="nofollow" class="external text" href="https://lirias.kuleuven.be/handle/123456789/136347">"Students' misconceptions of statistical inference: A review of the empirical evidence from research on statistics education"</a>. <i>Educational Research Review</i>. <b>2</b> (2): 98–113. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.edurev.2007.04.001">10.1016/j.edurev.2007.04.001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Educational+Research+Review&rft.atitle=Students%27+misconceptions+of+statistical+inference%3A+A+review+of+the+empirical+evidence+from+research+on+statistics+education&rft.volume=2&rft.issue=2&rft.pages=98-113&rft.date=2007&rft_id=info%3Adoi%2F10.1016%2Fj.edurev.2007.04.001&rft.aulast=Sotos&rft.aufirst=A.E.C.&rft.au=Vanhoof%2C+S.&rft.au=Van+den+Noortgate%2C+W.&rft.au=Onghena%2C+P.&rft_id=https%3A%2F%2Flirias.kuleuven.be%2Fhandle%2F123456789%2F136347&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20230602200310/https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/v/sampling-distribution-of-the-sample-mean">"Sampling distribution of the sample mean (video) | Khan Academy"</a>. 2 June 2023. Archived from <a rel="nofollow" class="external text" href="https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library/sample-means/v/sampling-distribution-of-the-sample-mean">the original</a> on 2023-06-02<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-10-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Sampling+distribution+of+the+sample+mean+%28video%29+%7C+Khan+Academy&rft.date=2023-06-02&rft_id=https%3A%2F%2Fwww.khanacademy.org%2Fmath%2Fstatistics-probability%2Fsampling-distributions-library%2Fsample-means%2Fv%2Fsampling-distribution-of-the-sample-mean&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRosenthal2000" class="citation book cs1">Rosenthal, Jeffrey Seth (2000). <i>A First Look at Rigorous Probability Theory</i>. World Scientific. Theorem 5.3.4, p. 47. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/981-02-4322-7" title="Special:BookSources/981-02-4322-7"><bdi>981-02-4322-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Look+at+Rigorous+Probability+Theory&rft.pages=Theorem+5.3.4%2C+p.+47&rft.pub=World+Scientific&rft.date=2000&rft.isbn=981-02-4322-7&rft.aulast=Rosenthal&rft.aufirst=Jeffrey+Seth&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohnson2004" class="citation book cs1">Johnson, Oliver Thomas (2004). <i>Information Theory and the Central Limit Theorem</i>. Imperial College Press. p. 88. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-86094-473-6" title="Special:BookSources/1-86094-473-6"><bdi>1-86094-473-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Information+Theory+and+the+Central+Limit+Theorem&rft.pages=88&rft.pub=Imperial+College+Press&rft.date=2004&rft.isbn=1-86094-473-6&rft.aulast=Johnson&rft.aufirst=Oliver+Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-Uchaikin-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-Uchaikin_28-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFUchaikinZolotarev1999" class="citation book cs1">Uchaikin, Vladimir V.; Zolotarev, V.M. (1999). <i>Chance and Stability: Stable distributions and their applications</i>. VSP. pp. 61–62. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/90-6764-301-7" title="Special:BookSources/90-6764-301-7"><bdi>90-6764-301-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chance+and+Stability%3A+Stable+distributions+and+their+applications&rft.pages=61-62&rft.pub=VSP&rft.date=1999&rft.isbn=90-6764-301-7&rft.aulast=Uchaikin&rft.aufirst=Vladimir+V.&rft.au=Zolotarev%2C+V.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorodinIbragimovSudakov1995" class="citation book cs1">Borodin, A. N.; Ibragimov, I. A.; Sudakov, V. N. (1995). <i>Limit Theorems for Functionals of Random Walks</i>. AMS Bookstore. Theorem 1.1, p. 8. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-0438-3" title="Special:BookSources/0-8218-0438-3"><bdi>0-8218-0438-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Limit+Theorems+for+Functionals+of+Random+Walks&rft.pages=Theorem+1.1%2C+p.+8&rft.pub=AMS+Bookstore&rft.date=1995&rft.isbn=0-8218-0438-3&rft.aulast=Borodin&rft.aufirst=A.+N.&rft.au=Ibragimov%2C+I.+A.&rft.au=Sudakov%2C+V.+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPetrov1976" class="citation book cs1">Petrov, V. V. (1976). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zSDqCAAAQBAJ"><i>Sums of Independent Random Variables</i></a>. New York-Heidelberg: Springer-Verlag. ch. 7. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783642658099" title="Special:BookSources/9783642658099"><bdi>9783642658099</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sums+of+Independent+Random+Variables&rft.place=New+York-Heidelberg&rft.pages=ch.+7&rft.pub=Springer-Verlag&rft.date=1976&rft.isbn=9783642658099&rft.aulast=Petrov&rft.aufirst=V.+V.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DzSDqCAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHew2017" class="citation journal cs1">Hew, Patrick Chisan (2017). "Asymptotic distribution of rewards accumulated by alternating renewal processes". <i>Statistics and Probability Letters</i>. <b>129</b>: 355–359. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.spl.2017.06.027">10.1016/j.spl.2017.06.027</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistics+and+Probability+Letters&rft.atitle=Asymptotic+distribution+of+rewards+accumulated+by+alternating+renewal+processes&rft.volume=129&rft.pages=355-359&rft.date=2017&rft_id=info%3Adoi%2F10.1016%2Fj.spl.2017.06.027&rft.aulast=Hew&rft.aufirst=Patrick+Chisan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-Rempala-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rempala_32-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRempalaWesolowski2002" class="citation journal cs1">Rempala, G.; Wesolowski, J. (2002). <a rel="nofollow" class="external text" href="https://projecteuclid.org/journals/electronic-communications-in-probability/volume-7/issue-none/Asymptotics-for-Products-of-Sums-and-U-statistics/10.1214/ECP.v7-1046.pdf">"Asymptotics of products of sums and <i>U</i>-statistics"</a> <span class="cs1-format">(PDF)</span>. <i>Electronic Communications in Probability</i>. <b>7</b>: 47–54. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Fecp.v7-1046">10.1214/ecp.v7-1046</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Electronic+Communications+in+Probability&rft.atitle=Asymptotics+of+products+of+sums+and+U-statistics&rft.volume=7&rft.pages=47-54&rft.date=2002&rft_id=info%3Adoi%2F10.1214%2Fecp.v7-1046&rft.aulast=Rempala&rft.aufirst=G.&rft.au=Wesolowski%2C+J.&rft_id=https%3A%2F%2Fprojecteuclid.org%2Fjournals%2Felectronic-communications-in-probability%2Fvolume-7%2Fissue-none%2FAsymptotics-for-Products-of-Sums-and-U-statistics%2F10.1214%2FECP.v7-1046.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEKlartag2007Theorem_1.2-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKlartag2007Theorem_1.2_33-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKlartag2007">Klartag (2007)</a>, Theorem 1.2.</span> </li> <li id="cite_note-FOOTNOTEDurrett2004Section_2.4,_Example_4.5-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDurrett2004Section_2.4,_Example_4.5_34-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDurrett2004">Durrett (2004)</a>, Section 2.4, Example 4.5.</span> </li> <li id="cite_note-FOOTNOTEKlartag2008Theorem_1-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKlartag2008Theorem_1_35-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKlartag2008">Klartag (2008)</a>, Theorem 1.</span> </li> <li id="cite_note-FOOTNOTEKlartag2007Theorem_1.1-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKlartag2007Theorem_1.1_36-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKlartag2007">Klartag (2007)</a>, Theorem 1.1.</span> </li> <li id="cite_note-Zygmund-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-Zygmund_37-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZygmund2003" class="citation book cs1"><a href="/wiki/Antoni_Zygmund" title="Antoni Zygmund">Zygmund, Antoni</a> (2003) [1959]. <a href="/wiki/Trigonometric_Series" title="Trigonometric Series"><i>Trigonometric Series</i></a>. Cambridge University Press. vol. II, sect. XVI.5, Theorem 5-5. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-89053-5" title="Special:BookSources/0-521-89053-5"><bdi>0-521-89053-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Trigonometric+Series&rft.pages=vol.+II%2C+sect.+XVI.5%2C+Theorem+5-5&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=0-521-89053-5&rft.aulast=Zygmund&rft.aufirst=Antoni&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEGaposhkin1966Theorem_2.1.13-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGaposhkin1966Theorem_2.1.13_38-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGaposhkin1966">Gaposhkin (1966)</a>, Theorem 2.1.13.</span> </li> <li id="cite_note-FOOTNOTEBárányVu2007Theorem_1.1-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBárányVu2007Theorem_1.1_39-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBárányVu2007">Bárány & Vu (2007)</a>, Theorem 1.1.</span> </li> <li id="cite_note-FOOTNOTEBárányVu2007Theorem_1.2-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBárányVu2007Theorem_1.2_40-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBárányVu2007">Bárány & Vu (2007)</a>, Theorem 1.2.</span> </li> <li id="cite_note-Meckes-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-Meckes_41-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMeckes2008" class="citation journal cs1"><a href="/wiki/Elizabeth_Meckes" title="Elizabeth Meckes">Meckes, Elizabeth</a> (2008). "Linear functions on the classical matrix groups". <i>Transactions of the American Mathematical Society</i>. <b>360</b> (10): 5355–5366. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0509441">math/0509441</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9947-08-04444-9">10.1090/S0002-9947-08-04444-9</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11981408">11981408</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=Linear+functions+on+the+classical+matrix+groups&rft.volume=360&rft.issue=10&rft.pages=5355-5366&rft.date=2008&rft_id=info%3Aarxiv%2Fmath%2F0509441&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A11981408%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1090%2FS0002-9947-08-04444-9&rft.aulast=Meckes&rft.aufirst=Elizabeth&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEGaposhkin1966Sect._1.5-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGaposhkin1966Sect._1.5_42-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGaposhkin1966">Gaposhkin (1966)</a>, Sect. 1.5.</span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKotaniSunada2003" class="citation book cs1">Kotani, M.; <a href="/wiki/Toshikazu_Sunada" title="Toshikazu Sunada">Sunada, Toshikazu</a> (2003). <i>Spectral geometry of crystal lattices</i>. Vol. 338. Contemporary Math. pp. 271–305. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4269-0" title="Special:BookSources/978-0-8218-4269-0"><bdi>978-0-8218-4269-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spectral+geometry+of+crystal+lattices&rft.pages=271-305&rft.pub=Contemporary+Math&rft.date=2003&rft.isbn=978-0-8218-4269-0&rft.aulast=Kotani&rft.aufirst=M.&rft.au=Sunada%2C+Toshikazu&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSunada2012" class="citation book cs1"><a href="/wiki/Toshikazu_Sunada" title="Toshikazu Sunada">Sunada, Toshikazu</a> (2012). <i>Topological Crystallography – With a View Towards Discrete Geometric Analysis</i>. Surveys and Tutorials in the Applied Mathematical Sciences. Vol. 6. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-4-431-54177-6" title="Special:BookSources/978-4-431-54177-6"><bdi>978-4-431-54177-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Crystallography+%E2%80%93+With+a+View+Towards+Discrete+Geometric+Analysis&rft.series=Surveys+and+Tutorials+in+the+Applied+Mathematical+Sciences&rft.pub=Springer&rft.date=2012&rft.isbn=978-4-431-54177-6&rft.aulast=Sunada&rft.aufirst=Toshikazu&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-Marasinghe-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-Marasinghe_45-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarasingheMeekerCookShin1994" class="citation conference cs1">Marasinghe, M.; Meeker, W.; Cook, D.; Shin, T. S. (August 1994). <i>Using graphics and simulation to teach statistical concepts</i>. Annual meeting of the American Statistician Association, Toronto, Canada.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Using+graphics+and+simulation+to+teach+statistical+concepts&rft.date=1994-08&rft.aulast=Marasinghe&rft.aufirst=M.&rft.au=Meeker%2C+W.&rft.au=Cook%2C+D.&rft.au=Shin%2C+T.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-Tijms-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-Tijms_46-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHenk2004" class="citation book cs1">Henk, Tijms (2004). <i>Understanding Probability: Chance Rules in Everyday Life</i>. Cambridge: Cambridge University Press. p. 169. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-54036-4" title="Special:BookSources/0-521-54036-4"><bdi>0-521-54036-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Understanding+Probability%3A+Chance+Rules+in+Everyday+Life&rft.place=Cambridge&rft.pages=169&rft.pub=Cambridge+University+Press&rft.date=2004&rft.isbn=0-521-54036-4&rft.aulast=Henk&rft.aufirst=Tijms&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGalton1889" class="citation book cs1">Galton, F. (1889). <a rel="nofollow" class="external text" href="http://galton.org/cgi-bin/searchImages/galton/search/books/natural-inheritance/pages/natural-inheritance_0073.htm"><i>Natural Inheritance</i></a>. p. 66.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Natural+Inheritance&rft.pages=66&rft.date=1889&rft.aulast=Galton&rft.aufirst=F.&rft_id=http%3A%2F%2Fgalton.org%2Fcgi-bin%2FsearchImages%2Fgalton%2Fsearch%2Fbooks%2Fnatural-inheritance%2Fpages%2Fnatural-inheritance_0073.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-Polya1920-48"><span class="mw-cite-backlink">^ <a href="#cite_ref-Polya1920_48-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Polya1920_48-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPólya1920" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/George_P%C3%B3lya" title="George Pólya">Pólya, George</a> (1920). <a rel="nofollow" class="external text" href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN266833020_0008">"Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und das Momentenproblem"</a> [On the central limit theorem of probability calculation and the problem of moments]. <i><a href="/wiki/Mathematische_Zeitschrift" title="Mathematische Zeitschrift">Mathematische Zeitschrift</a></i> (in German). <b>8</b> (3–4): 171–181. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01206525">10.1007/BF01206525</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123063388">123063388</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Zeitschrift&rft.atitle=%C3%9Cber+den+zentralen+Grenzwertsatz+der+Wahrscheinlichkeitsrechnung+und+das+Momentenproblem&rft.volume=8&rft.issue=3%E2%80%934&rft.pages=171-181&rft.date=1920&rft_id=info%3Adoi%2F10.1007%2FBF01206525&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123063388%23id-name%3DS2CID&rft.aulast=P%C3%B3lya&rft.aufirst=George&rft_id=http%3A%2F%2Fwww-gdz.sub.uni-goettingen.de%2Fcgi-bin%2Fdigbib.cgi%3FPPN266833020_0008&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-LC1986-49"><span class="mw-cite-backlink">^ <a href="#cite_ref-LC1986_49-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-LC1986_49-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-LC1986_49-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLe_Cam1986" class="citation journal cs1"><a href="/wiki/Lucien_Le_Cam" title="Lucien Le Cam">Le Cam, Lucien</a> (1986). <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.ss/1177013818">"The central limit theorem around 1935"</a>. <i>Statistical Science</i>. <b>1</b> (1): 78–91. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Fss%2F1177013818">10.1214/ss/1177013818</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistical+Science&rft.atitle=The+central+limit+theorem+around+1935&rft.volume=1&rft.issue=1&rft.pages=78-91&rft.date=1986&rft_id=info%3Adoi%2F10.1214%2Fss%2F1177013818&rft.aulast=Le+Cam&rft.aufirst=Lucien&rft_id=http%3A%2F%2Fprojecteuclid.org%2Feuclid.ss%2F1177013818&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-Hald-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hald_50-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHald1998" class="citation book cs1">Hald, Andreas (22 April 1998). <a rel="nofollow" class="external text" href="http://www.gbv.de/dms/goettingen/229762905.pdf"><i>A History of Mathematical Statistics from 1750 to 1930</i></a> <span class="cs1-format">(PDF)</span>. Wiley. chapter 17. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0471179122" title="Special:BookSources/978-0471179122"><bdi>978-0471179122</bdi></a>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://www.gbv.de/dms/goettingen/229762905.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-10-09.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematical+Statistics+from+1750+to+1930&rft.pages=chapter+17&rft.pub=Wiley&rft.date=1998-04-22&rft.isbn=978-0471179122&rft.aulast=Hald&rft.aufirst=Andreas&rft_id=http%3A%2F%2Fwww.gbv.de%2Fdms%2Fgoettingen%2F229762905.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEFischer2011Chapter_2;_Chapter_5.2-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFischer2011Chapter_2;_Chapter_5.2_51-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFischer2011">Fischer (2011)</a>, Chapter 2; Chapter 5.2.</span> </li> <li id="cite_note-Bernstein-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bernstein_52-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernstein1945" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Sergei_Natanovich_Bernstein" class="mw-redirect" title="Sergei Natanovich Bernstein">Bernstein, S. N.</a> (1945). "On the work of P. L. Chebyshev in Probability Theory". In Bernstein., S. N. (ed.). <i>Nauchnoe Nasledie P. L. Chebysheva. Vypusk Pervyi: Matematika</i> [<i>The Scientific Legacy of P. L. Chebyshev. Part I: Mathematics</i>] (in Russian). Moscow & Leningrad: Academiya Nauk SSSR. p. 174.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=On+the+work+of+P.+L.+Chebyshev+in+Probability+Theory&rft.btitle=Nauchnoe+Nasledie+P.+L.+Chebysheva.+Vypusk+Pervyi%3A+Matematika&rft.place=Moscow+%26+Leningrad&rft.pages=174&rft.pub=Academiya+Nauk+SSSR&rft.date=1945&rft.aulast=Bernstein&rft.aufirst=S.+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZabell1995" class="citation journal cs1">Zabell, S. L. (1995). "Alan Turing and the Central Limit Theorem". <i>American Mathematical Monthly</i>. <b>102</b> (6): 483–494. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00029890.1995.12004608">10.1080/00029890.1995.12004608</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Alan+Turing+and+the+Central+Limit+Theorem&rft.volume=102&rft.issue=6&rft.pages=483-494&rft.date=1995&rft_id=info%3Adoi%2F10.1080%2F00029890.1995.12004608&rft.aulast=Zabell&rft.aufirst=S.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> <li id="cite_note-Jørgensen-1997-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-Jørgensen-1997_54-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJørgensen1997" class="citation book cs1">Jørgensen, Bent (1997). <i>The Theory of Dispersion Models</i>. Chapman & Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0412997112" title="Special:BookSources/978-0412997112"><bdi>978-0412997112</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Dispersion+Models&rft.pub=Chapman+%26+Hall&rft.date=1997&rft.isbn=978-0412997112&rft.aulast=J%C3%B8rgensen&rft.aufirst=Bent&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=34" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBárányVu2007" class="citation journal cs1"><a href="/wiki/Imre_B%C3%A1r%C3%A1ny" title="Imre Bárány">Bárány, Imre</a>; Vu, Van (2007). "Central limit theorems for Gaussian polytopes". <i>Annals of Probability</i>. <b>35</b> (4). Institute of Mathematical Statistics: 1593–1621. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0610192">math/0610192</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2F009117906000000791">10.1214/009117906000000791</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:9128253">9128253</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Probability&rft.atitle=Central+limit+theorems+for+Gaussian+polytopes&rft.volume=35&rft.issue=4&rft.pages=1593-1621&rft.date=2007&rft_id=info%3Aarxiv%2Fmath%2F0610192&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A9128253%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1214%2F009117906000000791&rft.aulast=B%C3%A1r%C3%A1ny&rft.aufirst=Imre&rft.au=Vu%2C+Van&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBauer2001" class="citation book cs1">Bauer, Heinz (2001). <i>Measure and Integration Theory</i>. Berlin: de Gruyter. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3110167190" title="Special:BookSources/3110167190"><bdi>3110167190</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Measure+and+Integration+Theory&rft.place=Berlin&rft.pub=de+Gruyter&rft.date=2001&rft.isbn=3110167190&rft.aulast=Bauer&rft.aufirst=Heinz&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBillingsley1995" class="citation book cs1">Billingsley, Patrick (1995). <i>Probability and Measure</i> (3rd ed.). John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-00710-2" title="Special:BookSources/0-471-00710-2"><bdi>0-471-00710-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability+and+Measure&rft.edition=3rd&rft.pub=John+Wiley+%26+Sons&rft.date=1995&rft.isbn=0-471-00710-2&rft.aulast=Billingsley&rft.aufirst=Patrick&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBradley2005" class="citation journal cs1">Bradley, Richard (2005). "Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions". <i>Probability Surveys</i>. <b>2</b>: 107–144. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0511078">math/0511078</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005math.....11078B">2005math.....11078B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2F154957805100000104">10.1214/154957805100000104</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:8395267">8395267</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Probability+Surveys&rft.atitle=Basic+Properties+of+Strong+Mixing+Conditions.+A+Survey+and+Some+Open+Questions&rft.volume=2&rft.pages=107-144&rft.date=2005&rft_id=info%3Aarxiv%2Fmath%2F0511078&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A8395267%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1214%2F154957805100000104&rft_id=info%3Abibcode%2F2005math.....11078B&rft.aulast=Bradley&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBradley2007" class="citation book cs1">Bradley, Richard (2007). <i>Introduction to Strong Mixing Conditions</i> (1st ed.). Heber City, UT: Kendrick Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-9740427-9-4" title="Special:BookSources/978-0-9740427-9-4"><bdi>978-0-9740427-9-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Strong+Mixing+Conditions&rft.place=Heber+City%2C+UT&rft.edition=1st&rft.pub=Kendrick+Press&rft.date=2007&rft.isbn=978-0-9740427-9-4&rft.aulast=Bradley&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDinovChristouSanchez2008" class="citation journal cs1">Dinov, Ivo; Christou, Nicolas; Sanchez, Juana (2008). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160303185802/http://www.amstat.org/publications/jse/v16n2/dinov.html">"Central Limit Theorem: New SOCR Applet and Demonstration Activity"</a>. <i>Journal of Statistics Education</i>. <b>16</b> (2). ASA: 1–15. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F10691898.2008.11889560">10.1080/10691898.2008.11889560</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3152447">3152447</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/21833159">21833159</a>. Archived from <a rel="nofollow" class="external text" href="http://www.amstat.org/publications/jse/v16n2/dinov.html">the original</a> on 2016-03-03<span class="reference-accessdate">. Retrieved <span class="nowrap">2008-08-23</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Statistics+Education&rft.atitle=Central+Limit+Theorem%3A+New+SOCR+Applet+and+Demonstration+Activity&rft.volume=16&rft.issue=2&rft.pages=1-15&rft.date=2008&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3152447%23id-name%3DPMC&rft_id=info%3Apmid%2F21833159&rft_id=info%3Adoi%2F10.1080%2F10691898.2008.11889560&rft.aulast=Dinov&rft.aufirst=Ivo&rft.au=Christou%2C+Nicolas&rft.au=Sanchez%2C+Juana&rft_id=http%3A%2F%2Fwww.amstat.org%2Fpublications%2Fjse%2Fv16n2%2Fdinov.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDurrett2004" class="citation book cs1"><a href="/wiki/Rick_Durrett" title="Rick Durrett">Durrett, Richard</a> (2004). <i>Probability: theory and examples</i> (3rd ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0521765390" title="Special:BookSources/0521765390"><bdi>0521765390</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability%3A+theory+and+examples&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=2004&rft.isbn=0521765390&rft.aulast=Durrett&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFischer2011" class="citation book cs1">Fischer, Hans (2011). <a rel="nofollow" class="external text" href="http://www.medicine.mcgill.ca/epidemiology/hanley/bios601/GaussianModel/HistoryCentralLimitTheorem.pdf"><i>A History of the Central Limit Theorem: From Classical to Modern Probability Theory</i></a> <span class="cs1-format">(PDF)</span>. Sources and Studies in the History of Mathematics and Physical Sciences. New York: Springer. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-387-87857-7">10.1007/978-0-387-87857-7</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-87856-0" title="Special:BookSources/978-0-387-87856-0"><bdi>978-0-387-87856-0</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2743162">2743162</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1226.60004">1226.60004</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171031171033/http://www.medicine.mcgill.ca/epidemiology/hanley/bios601/GaussianModel/HistoryCentralLimitTheorem.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2017-10-31.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+the+Central+Limit+Theorem%3A+From+Classical+to+Modern+Probability+Theory&rft.place=New+York&rft.series=Sources+and+Studies+in+the+History+of+Mathematics+and+Physical+Sciences&rft.pub=Springer&rft.date=2011&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1226.60004%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2743162%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-0-387-87857-7&rft.isbn=978-0-387-87856-0&rft.aulast=Fischer&rft.aufirst=Hans&rft_id=http%3A%2F%2Fwww.medicine.mcgill.ca%2Fepidemiology%2Fhanley%2Fbios601%2FGaussianModel%2FHistoryCentralLimitTheorem.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGaposhkin1966" class="citation journal cs1">Gaposhkin, V. F. (1966). "Lacunary series and independent functions". <i>Russian Mathematical Surveys</i>. <b>21</b> (6): 1–82. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1966RuMaS..21....1G">1966RuMaS..21....1G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1070%2FRM1966v021n06ABEH001196">10.1070/RM1966v021n06ABEH001196</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:250833638">250833638</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Russian+Mathematical+Surveys&rft.atitle=Lacunary+series+and+independent+functions&rft.volume=21&rft.issue=6&rft.pages=1-82&rft.date=1966&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A250833638%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1070%2FRM1966v021n06ABEH001196&rft_id=info%3Abibcode%2F1966RuMaS..21....1G&rft.aulast=Gaposhkin&rft.aufirst=V.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlartag2007" class="citation journal cs1">Klartag, Bo'az (2007). "A central limit theorem for convex sets". <i>Inventiones Mathematicae</i>. <b>168</b> (1): 91–131. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0605014">math/0605014</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007InMat.168...91K">2007InMat.168...91K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00222-006-0028-8">10.1007/s00222-006-0028-8</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119169773">119169773</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Inventiones+Mathematicae&rft.atitle=A+central+limit+theorem+for+convex+sets&rft.volume=168&rft.issue=1&rft.pages=91-131&rft.date=2007&rft_id=info%3Aarxiv%2Fmath%2F0605014&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119169773%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00222-006-0028-8&rft_id=info%3Abibcode%2F2007InMat.168...91K&rft.aulast=Klartag&rft.aufirst=Bo%27az&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlartag2008" class="citation journal cs1">Klartag, Bo'az (2008). "A Berry–Esseen type inequality for convex bodies with an unconditional basis". <i>Probability Theory and Related Fields</i>. <b>145</b> (1–2): 1–33. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0705.0832">0705.0832</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00440-008-0158-6">10.1007/s00440-008-0158-6</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:10163322">10163322</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Probability+Theory+and+Related+Fields&rft.atitle=A+Berry%E2%80%93Esseen+type+inequality+for+convex+bodies+with+an+unconditional+basis&rft.volume=145&rft.issue=1%E2%80%932&rft.pages=1-33&rft.date=2008&rft_id=info%3Aarxiv%2F0705.0832&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A10163322%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00440-008-0158-6&rft.aulast=Klartag&rft.aufirst=Bo%27az&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Central_limit_theorem&action=edit&section=35" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; 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Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/CentralLimitTheorem.html">"Central Limit Theorem"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Central+Limit+Theorem&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FCentralLimitTheorem.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentral+limit+theorem" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="https://www.mctague.org/carl/blog/2021/04/23/central-limit-theorem/">A music video demonstrating the central limit theorem with a Galton board</a> by Carl McTague</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist 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navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Descriptive_statistics" style="font-size:114%;margin:0 4em"><a href="/wiki/Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">Continuous data</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Central_tendency" title="Central tendency">Center</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mean" title="Mean">Mean</a> <ul><li><a href="/wiki/Arithmetic_mean" title="Arithmetic mean">Arithmetic</a></li> <li><a href="/wiki/Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic-Geometric</a></li> <li><a href="/wiki/Contraharmonic_mean" title="Contraharmonic mean">Contraharmonic</a></li> <li><a href="/wiki/Cubic_mean" title="Cubic mean">Cubic</a></li> <li><a href="/wiki/Generalized_mean" title="Generalized mean">Generalized/power</a></li> <li><a href="/wiki/Geometric_mean" title="Geometric mean">Geometric</a></li> <li><a href="/wiki/Harmonic_mean" title="Harmonic mean">Harmonic</a></li> <li><a href="/wiki/Heronian_mean" title="Heronian mean">Heronian</a></li> <li><a href="/wiki/Heinz_mean" title="Heinz mean">Heinz</a></li> <li><a href="/wiki/Lehmer_mean" title="Lehmer mean">Lehmer</a></li></ul></li> <li><a href="/wiki/Median" title="Median">Median</a></li> <li><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_dispersion" title="Statistical dispersion">Dispersion</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Average_absolute_deviation" title="Average absolute deviation">Average absolute deviation</a></li> <li><a href="/wiki/Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li> <li><a href="/wiki/Interquartile_range" title="Interquartile range">Interquartile range</a></li> <li><a href="/wiki/Percentile" title="Percentile">Percentile</a></li> <li><a href="/wiki/Range_(statistics)" title="Range (statistics)">Range</a></li> <li><a href="/wiki/Standard_deviation" title="Standard deviation">Standard deviation</a></li> <li><a href="/wiki/Variance#Sample_variance" title="Variance">Variance</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Shape_of_the_distribution" class="mw-redirect" title="Shape of the distribution">Shape</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Central limit theorem</a></li> <li><a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">Moments</a> <ul><li><a href="/wiki/Kurtosis" title="Kurtosis">Kurtosis</a></li> <li><a href="/wiki/L-moment" title="L-moment">L-moments</a></li> <li><a href="/wiki/Skewness" title="Skewness">Skewness</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Count_data" title="Count data">Count data</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Summary tables</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li> <li><a href="/wiki/Grouped_data" title="Grouped data">Grouped data</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Dependence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson product-moment correlation</a></li> <li><a href="/wiki/Rank_correlation" title="Rank correlation">Rank correlation</a> <ul><li><a href="/wiki/Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li> <li><a href="/wiki/Spearman%27s_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li></ul></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_graphics" title="Statistical graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bar_chart" title="Bar chart">Bar chart</a></li> <li><a href="/wiki/Biplot" title="Biplot">Biplot</a></li> <li><a href="/wiki/Box_plot" title="Box plot">Box plot</a></li> <li><a href="/wiki/Control_chart" title="Control chart">Control chart</a></li> <li><a href="/wiki/Correlogram" title="Correlogram">Correlogram</a></li> <li><a href="/wiki/Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li> <li><a href="/wiki/Forest_plot" title="Forest plot">Forest plot</a></li> <li><a href="/wiki/Histogram" title="Histogram">Histogram</a></li> <li><a href="/wiki/Pie_chart" title="Pie chart">Pie chart</a></li> <li><a href="/wiki/Q%E2%80%93Q_plot" title="Q–Q plot">Q–Q plot</a></li> <li><a href="/wiki/Radar_chart" title="Radar chart">Radar chart</a></li> <li><a href="/wiki/Run_chart" title="Run chart">Run chart</a></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li> <li><a href="/wiki/Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li> <li><a href="/wiki/Violin_plot" title="Violin plot">Violin plot</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_collection" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_collection" title="Data collection">Data collection</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Design_of_experiments" title="Design of experiments">Study design</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Effect_size" title="Effect size">Effect size</a></li> <li><a href="/wiki/Missing_data" title="Missing data">Missing data</a></li> <li><a href="/wiki/Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li> <li><a href="/wiki/Statistical_population" title="Statistical population">Population</a></li> <li><a href="/wiki/Replication_(statistics)" title="Replication (statistics)">Replication</a></li> <li><a href="/wiki/Sample_size_determination" title="Sample size determination">Sample size determination</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survey_methodology" title="Survey methodology">Survey methodology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">Sampling</a> <ul><li><a href="/wiki/Cluster_sampling" title="Cluster sampling">Cluster</a></li> <li><a href="/wiki/Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li> <li><a href="/wiki/Opinion_poll" title="Opinion poll">Opinion poll</a></li> <li><a href="/wiki/Questionnaire" title="Questionnaire">Questionnaire</a></li> <li><a href="/wiki/Standard_error" title="Standard error">Standard error</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Experiment" title="Experiment">Controlled experiments</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li> <li><a href="/wiki/Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li> <li><a href="/wiki/Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li> <li><a href="/wiki/Random_assignment" title="Random assignment">Random assignment</a></li> <li><a href="/wiki/Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li> <li><a href="/wiki/Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li> <li><a href="/wiki/Scientific_control" title="Scientific control">Scientific control</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive designs</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li> <li><a href="/wiki/Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li> <li><a href="/wiki/Up-and-Down_Designs" class="mw-redirect" title="Up-and-Down Designs">Up-and-down designs</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Observational_study" title="Observational study">Observational studies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohort_study" title="Cohort study">Cohort study</a></li> <li><a href="/wiki/Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li> <li><a href="/wiki/Natural_experiment" title="Natural experiment">Natural experiment</a></li> <li><a href="/wiki/Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a></li> <li><a href="/wiki/Sampling_distribution" title="Sampling distribution">Sampling distribution</a> <ul><li><a href="/wiki/Order_statistic" title="Order statistic">Order statistic</a></li></ul></li> <li><a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a> <ul><li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li></ul></li> <li><a href="/wiki/Statistical_model" title="Statistical model">Statistical model</a> <ul><li><a href="/wiki/Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li> <li><a href="/wiki/Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li> <li><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameter</a> <ul><li><a href="/wiki/Location_parameter" title="Location parameter">location</a></li> <li><a href="/wiki/Scale_parameter" title="Scale parameter">scale</a></li> <li><a href="/wiki/Shape_parameter" title="Shape parameter">shape</a></li></ul></li> <li><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric family</a> <ul><li><a href="/wiki/Likelihood_function" title="Likelihood function">Likelihood</a> <a href="/wiki/Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size:85%;">(monotone)</span></a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential family</a></li></ul></li> <li><a href="/wiki/Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li> <li><a href="/wiki/Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/U-statistic" title="U-statistic">U</a></li> <li><a href="/wiki/V-statistic" title="V-statistic">V</a></li></ul></li> <li><a href="/wiki/Optimal_decision" title="Optimal decision">Optimal decision</a> <ul><li><a href="/wiki/Loss_function" title="Loss function">loss function</a></li></ul></li> <li><a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li> <li><a href="/wiki/Statistical_distance" title="Statistical distance">Statistical distance</a> <ul><li><a href="/wiki/Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li> <li><a href="/wiki/Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li> <li><a href="/wiki/Robust_statistics" title="Robust statistics">Robustness</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Point_estimation" title="Point estimation">Point estimation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Estimating_equations" title="Estimating equations">Estimating equations</a> <ul><li><a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li> <li><a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li> <li><a href="/wiki/M-estimator" title="M-estimator">M-estimator</a></li> <li><a href="/wiki/Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li> <li><a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a> <ul><li><a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a> <ul><li><a href="/wiki/Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li> <li><a href="/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li> <li><a href="/wiki/Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Plug-in</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Interval_estimation" title="Interval estimation">Interval estimation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Confidence_interval" title="Confidence interval">Confidence interval</a></li> <li><a href="/wiki/Pivotal_quantity" title="Pivotal quantity">Pivot</a></li> <li><a href="/wiki/Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li> <li><a href="/wiki/Prediction_interval" title="Prediction interval">Prediction interval</a></li> <li><a href="/wiki/Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li> <li><a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">Resampling</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Testing hypotheses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li> <li><a href="/wiki/Power_(statistics)" title="Power (statistics)">Power</a> <ul><li><a href="/wiki/Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li> <li><a href="/wiki/Permutation_test" title="Permutation test">Permutation test</a> <ul><li><a href="/wiki/Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li> <li><a href="/wiki/Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li> <li><a href="/wiki/Score_test" title="Score test">Score/Lagrange multiplier</a></li> <li><a href="/wiki/Wald_test" title="Wald test">Wald</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/List_of_statistical_tests" title="List of statistical tests">Specific tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Z-test" title="Z-test"><i>Z</i>-test <span style="font-size:85%;">(normal)</span></a></li> <li><a href="/wiki/Student%27s_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li> <li><a href="/wiki/F-test" title="F-test"><i>F</i>-test</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chi-squared_test" title="Chi-squared test">Chi-squared</a></li> <li><a href="/wiki/G-test" title="G-test"><i>G</i>-test</a></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li> <li><a href="/wiki/Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li> <li><a href="/wiki/Lilliefors_test" title="Lilliefors test">Lilliefors</a></li> <li><a href="/wiki/Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li> <li><a href="/wiki/Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size:85%;">(Shapiro–Wilk)</span></a></li> <li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li> <li><a href="/wiki/Model_selection" title="Model selection">Model selection</a> <ul><li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li> <li><a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">AIC</a></li> <li><a href="/wiki/Bayesian_information_criterion" title="Bayesian information criterion">BIC</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sign_test" title="Sign test">Sign</a> <ul><li><a href="/wiki/Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li> <li><a href="/wiki/Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size:85%;">(Wilcoxon)</span></a> <ul><li><a href="/wiki/Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li> <li><a href="/wiki/Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size:85%;">(Mann–Whitney)</span></a></li> <li><a href="/wiki/Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="/wiki/Analysis_of_variance" title="Analysis of variance">anova</a> <ul><li><a href="/wiki/Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size:85%;">(Kruskal–Wallis)</span></a></li> <li><a href="/wiki/Friedman_test" title="Friedman test">2-way <span style="font-size:85%;">(Friedman)</span></a></li> <li><a href="/wiki/Jonckheere%27s_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size:85%;">(Jonckheere–Terpstra)</span></a></li></ul></li> <li><a href="/wiki/Van_der_Waerden_test" title="Van der Waerden test">Van der Waerden test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian probability</a> <ul><li><a href="/wiki/Prior_probability" title="Prior probability">prior</a></li> <li><a href="/wiki/Posterior_probability" title="Posterior probability">posterior</a></li></ul></li> <li><a href="/wiki/Credible_interval" title="Credible interval">Credible interval</a></li> <li><a href="/wiki/Bayes_factor" title="Bayes factor">Bayes factor</a></li> <li><a href="/wiki/Bayes_estimator" title="Bayes estimator">Bayesian estimator</a> <ul><li><a href="/wiki/Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="CorrelationRegression_analysis" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></li></ul></div></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li> <li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Confounding" title="Confounding">Confounding variable</a></li> <li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li> <li><a href="/wiki/Regression_validation" title="Regression validation">Regression validation</a></li> <li><a href="/wiki/Mixed_model" title="Mixed model">Mixed effects models</a></li> <li><a href="/wiki/Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li> <li><a href="/wiki/Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">Multivariate adaptive regression splines (MARS)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li> <li><a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li> <li><a href="/wiki/General_linear_model" title="General linear model">General linear model</a></li> <li><a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Bayesian regression</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Non-standard predictors</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li> <li><a href="/wiki/Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li> <li><a href="/wiki/Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li> <li><a href="/wiki/Isotonic_regression" title="Isotonic regression">Isotonic</a></li> <li><a href="/wiki/Robust_regression" title="Robust regression">Robust</a></li> <li><a href="/wiki/Heteroscedasticity" class="mw-redirect" title="Heteroscedasticity">Heteroscedasticity</a></li> <li><a href="/wiki/Homoscedasticity" class="mw-redirect" title="Homoscedasticity">Homoscedasticity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Exponential_family" title="Exponential family">Exponential families</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic <span style="font-size:85%;">(Bernoulli)</span></a> / <a href="/wiki/Binomial_regression" title="Binomial regression">Binomial</a> / <a href="/wiki/Poisson_regression" title="Poisson regression">Poisson regressions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Partition_of_sums_of_squares" title="Partition of sums of squares">Partition of variance</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li> <li><a href="/wiki/Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li> <li><a href="/wiki/Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">Degrees of freedom</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_Multivariate_/_Time-series_/_Survival_analysis" style="font-size:114%;margin:0 4em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a> / <a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a> / <a href="/wiki/Time_series" title="Time series">Time-series</a> / <a href="/wiki/Survival_analysis" title="Survival analysis">Survival analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohen%27s_kappa" title="Cohen's kappa">Cohen's kappa</a></li> <li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Graphical_model" title="Graphical model">Graphical model</a></li> <li><a href="/wiki/Poisson_regression" title="Poisson regression">Log-linear model</a></li> <li><a href="/wiki/McNemar%27s_test" title="McNemar's test">McNemar's test</a></li> <li><a href="/wiki/Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics" title="Cochran–Mantel–Haenszel statistics">Cochran–Mantel–Haenszel statistics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_linear_model" title="General linear model">Regression</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li> <li><a href="/wiki/Principal_component_analysis" title="Principal component analysis">Principal components</a></li> <li><a href="/wiki/Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li> <li><a href="/wiki/Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li> <li><a href="/wiki/Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li> <li><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></li> <li><a href="/wiki/Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a> <ul><li><a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li> <li><a href="/wiki/Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a> <ul><li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a> <ul><li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Normal</a></li></ul></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Time_series" title="Time series">Time-series</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li> <li><a href="/wiki/Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationarity</a></li> <li><a href="/wiki/Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li> <li><a href="/wiki/Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li> <li><a href="/wiki/Cointegration" title="Cointegration">Cointegration</a></li> <li><a href="/wiki/Structural_break" title="Structural break">Structural break</a></li> <li><a href="/wiki/Granger_causality" title="Granger causality">Granger causality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Specific tests</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li> <li><a href="/wiki/Johansen_test" title="Johansen test">Johansen</a></li> <li><a href="/wiki/Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size:85%;">(Ljung–Box)</span></a></li> <li><a href="/wiki/Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li> <li><a href="/wiki/Breusch%E2%80%93Godfrey_test" title="Breusch–Godfrey test">Breusch–Godfrey</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Time_domain" title="Time domain">Time domain</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a> <ul><li><a href="/wiki/Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li> <li><a href="/wiki/Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li> <li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li> <li><a href="/wiki/Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size:85%;">(Box–Jenkins)</span></a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li> <li><a href="/wiki/Vector_autoregression" title="Vector autoregression">Vector autoregression (VAR)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Frequency_domain" title="Frequency domain">Frequency domain</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li> <li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li> <li><a href="/wiki/Wavelet" title="Wavelet">Wavelet</a></li> <li><a href="/wiki/Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survival_analysis" title="Survival analysis">Survival</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Survival_function" title="Survival function">Survival function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator (product limit)</a></li> <li><a href="/wiki/Proportional_hazards_model" title="Proportional hazards model">Proportional hazards models</a></li> <li><a href="/wiki/Accelerated_failure_time_model" title="Accelerated failure time model">Accelerated failure time (AFT) model</a></li> <li><a href="/wiki/First-hitting-time_model" title="First-hitting-time model">First hitting time</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Failure_rate" title="Failure rate">Hazard function</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nelson%E2%80%93Aalen_estimator" title="Nelson–Aalen estimator">Nelson–Aalen estimator</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Test</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Log-rank_test" class="mw-redirect" title="Log-rank test">Log-rank test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Applications" style="font-size:114%;margin:0 4em"><a href="/wiki/List_of_fields_of_application_of_statistics" title="List of fields of application of statistics">Applications</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Biostatistics" title="Biostatistics">Biostatistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bioinformatics" title="Bioinformatics">Bioinformatics</a></li> <li><a href="/wiki/Clinical_trial" title="Clinical trial">Clinical trials</a> / <a href="/wiki/Clinical_study_design" title="Clinical study design">studies</a></li> <li><a href="/wiki/Epidemiology" title="Epidemiology">Epidemiology</a></li> <li><a href="/wiki/Medical_statistics" title="Medical statistics">Medical statistics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Engineering_statistics" title="Engineering statistics">Engineering statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chemometrics" title="Chemometrics">Chemometrics</a></li> <li><a href="/wiki/Methods_engineering" title="Methods engineering">Methods engineering</a></li> <li><a href="/wiki/Probabilistic_design" title="Probabilistic design">Probabilistic design</a></li> <li><a href="/wiki/Statistical_process_control" title="Statistical process control">Process</a> / <a href="/wiki/Quality_control" title="Quality control">quality control</a></li> <li><a href="/wiki/Reliability_engineering" title="Reliability engineering">Reliability</a></li> <li><a href="/wiki/System_identification" title="System identification">System identification</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Social_statistics" title="Social statistics">Social statistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Actuarial_science" title="Actuarial science">Actuarial science</a></li> <li><a href="/wiki/Census" title="Census">Census</a></li> <li><a href="/wiki/Crime_statistics" title="Crime statistics">Crime statistics</a></li> <li><a href="/wiki/Demographic_statistics" title="Demographic statistics">Demography</a></li> <li><a href="/wiki/Econometrics" title="Econometrics">Econometrics</a></li> <li><a href="/wiki/Jurimetrics" title="Jurimetrics">Jurimetrics</a></li> <li><a href="/wiki/National_accounts" title="National accounts">National accounts</a></li> <li><a href="/wiki/Official_statistics" title="Official statistics">Official statistics</a></li> <li><a href="/wiki/Population_statistics" class="mw-redirect" title="Population statistics">Population statistics</a></li> <li><a href="/wiki/Psychometrics" title="Psychometrics">Psychometrics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Spatial_analysis" title="Spatial analysis">Spatial statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cartography" title="Cartography">Cartography</a></li> <li><a href="/wiki/Environmental_statistics" title="Environmental statistics">Environmental statistics</a></li> <li><a href="/wiki/Geographic_information_system" title="Geographic information system">Geographic information system</a></li> <li><a href="/wiki/Geostatistics" title="Geostatistics">Geostatistics</a></li> <li><a href="/wiki/Kriging" title="Kriging">Kriging</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" 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Central limit theorem - Wikipedia