Law of large numbers

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<span>Limitation</span> </div> </a> <ul id="toc-Limitation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Forms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Forms"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Forms</span> </div> </a> <button aria-controls="toc-Forms-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 Forms subsection</span> </button> <ul id="toc-Forms-sublist" class="vector-toc-list"> <li id="toc-Weak_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weak_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Weak law</span> </div> </a> <ul id="toc-Weak_law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Strong_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Strong_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Strong law</span> </div> </a> <ul id="toc-Strong_law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differences_between_the_weak_law_and_the_strong_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differences_between_the_weak_law_and_the_strong_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Differences between the weak law and the strong law</span> </div> </a> <ul id="toc-Differences_between_the_weak_law_and_the_strong_law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uniform_laws_of_large_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uniform_laws_of_large_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Uniform laws of large numbers</span> </div> </a> <ul id="toc-Uniform_laws_of_large_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Borel's_law_of_large_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Borel's_law_of_large_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Borel's law of large numbers</span> </div> </a> <ul id="toc-Borel's_law_of_large_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Proof_of_the_weak_law" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proof_of_the_weak_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Proof of the weak law</span> </div> </a> <button aria-controls="toc-Proof_of_the_weak_law-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 Proof of the weak law subsection</span> </button> <ul id="toc-Proof_of_the_weak_law-sublist" class="vector-toc-list"> <li id="toc-Proof_using_Chebyshev's_inequality_assuming_finite_variance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proof_using_Chebyshev's_inequality_assuming_finite_variance"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Proof using Chebyshev's inequality assuming finite variance</span> </div> </a> <ul id="toc-Proof_using_Chebyshev's_inequality_assuming_finite_variance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proof_using_convergence_of_characteristic_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proof_using_convergence_of_characteristic_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Proof using convergence of characteristic functions</span> </div> </a> <ul id="toc-Proof_using_convergence_of_characteristic_functions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Proof_of_the_strong_law" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proof_of_the_strong_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Proof of the strong law</span> </div> </a> <ul id="toc-Proof_of_the_strong_law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Consequences" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Consequences"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Consequences</span> </div> </a> <ul id="toc-Consequences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <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 vector-toc-list-item-expanded"> <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 vector-toc-list-item-expanded"> <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 vector-toc-list-item-expanded"> <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">Law of large numbers</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 51 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-51" 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">51 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%A7%D9%86%D9%88%D9%86_%D8%A7%D9%84%D8%A3%D8%B9%D8%AF%D8%A7%D8%AF_%D8%A7%D9%84%D9%83%D8%A8%D9%8A%D8%B1%D8%A9" title="قانون الأعداد الكبيرة – Arabic" lang="ar" hreflang="ar" data-title="قانون الأعداد الكبيرة" 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/Llei_de_los_grandes_n%C3%BAmberos" title="Llei de los grandes númberos – Asturian" lang="ast" hreflang="ast" data-title="Llei de los grandes númberos" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/B%C3%B6y%C3%BCk_%C9%99d%C9%99dl%C9%99r_qanunu" title="Böyük ədədlər qanunu – Azerbaijani" lang="az" hreflang="az" data-title="Böyük ədədlər qanunu" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%B2%D1%8F%D0%BB%D1%96%D0%BA%D1%96%D1%85_%D0%BB%D1%96%D0%BA%D0%B0%D1%9E" title="Закон вялікіх лікаў – Belarusian" lang="be" hreflang="be" data-title="Закон вялікіх лікаў" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%B7%D0%B0_%D0%B3%D0%BE%D0%BB%D0%B5%D0%BC%D0%B8%D1%82%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Закон за големите числа – Bulgarian" lang="bg" hreflang="bg" data-title="Закон за големите числа" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Llei_dels_grans_nombres" title="Llei dels grans nombres – Catalan" lang="ca" hreflang="ca" data-title="Llei dels grans nombres" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9F%D1%8B%D1%81%C4%83%D0%BA_%D1%85%D0%B8%D1%81%D0%B5%D0%BF%D1%81%D0%B5%D0%BD_%D1%81%D0%B0%D0%BA%D0%BA%D1%83%D0%BD%C4%95" title="Пысăк хисепсен саккунĕ – Chuvash" lang="cv" hreflang="cv" data-title="Пысăк хисепсен саккунĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Z%C3%A1kon_velk%C3%BDch_%C4%8D%C3%ADsel" title="Zákon velkých čísel – Czech" lang="cs" hreflang="cs" data-title="Zákon velkých čísel" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Deddf_niferoedd_mawr" title="Deddf niferoedd mawr – Welsh" lang="cy" hreflang="cy" data-title="Deddf niferoedd mawr" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Store_tals_lov" title="Store tals lov – Danish" lang="da" hreflang="da" data-title="Store tals lov" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gesetz_der_gro%C3%9Fen_Zahlen" title="Gesetz der großen Zahlen – German" lang="de" hreflang="de" data-title="Gesetz der großen Zahlen" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9D%CF%8C%CE%BC%CE%BF%CF%82_%CF%84%CF%89%CE%BD_%CE%BC%CE%B5%CE%B3%CE%AC%CE%BB%CF%89%CE%BD_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8E%CE%BD" title="Νόμος των μεγάλων αριθμών – Greek" lang="el" hreflang="el" data-title="Νόμος των μεγάλων αριθμών" 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/Ley_de_los_grandes_n%C3%BAmeros" title="Ley de los grandes números – Spanish" lang="es" hreflang="es" data-title="Ley de los grandes números" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Le%C4%9Do_de_grandaj_nombroj" title="Leĝo de grandaj nombroj – Esperanto" lang="eo" hreflang="eo" data-title="Leĝo de grandaj nombroj" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_handien_lege" title="Zenbaki handien lege – Basque" lang="eu" hreflang="eu" data-title="Zenbaki handien lege" 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%A7%D9%86%D9%88%D9%86_%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF_%D8%A8%D8%B2%D8%B1%DA%AF" title="قانون اعداد بزرگ – Persian" lang="fa" hreflang="fa" data-title="قانون اعداد بزرگ" 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/Loi_des_grands_nombres" title="Loi des grands nombres – French" lang="fr" hreflang="fr" data-title="Loi des grands nombres" 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/Lei_dos_grandes_n%C3%BAmeros" title="Lei dos grandes números – Galician" lang="gl" hreflang="gl" data-title="Lei dos grandes números" 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/%ED%81%B0_%EC%88%98%EC%9D%98_%EB%B2%95%EC%B9%99" title="큰 수의 법칙 – Korean" lang="ko" hreflang="ko" data-title="큰 수의 법칙" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A5%D5%AE_%D5%A9%D5%BE%D5%A5%D6%80%D5%AB_%D6%85%D6%80%D5%A5%D5%B6%D6%84" title="Մեծ թվերի օրենք – Armenian" lang="hy" hreflang="hy" data-title="Մեծ թվերի օրենք" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Zakon_velikih_brojeva" title="Zakon velikih brojeva – Croatian" lang="hr" hreflang="hr" data-title="Zakon velikih brojeva" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Legge_dei_grandi_numeri" title="Legge dei grandi numeri – Italian" lang="it" hreflang="it" data-title="Legge dei grandi numeri" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%95%D7%A7_%D7%94%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D_%D7%94%D7%92%D7%93%D7%95%D7%9C%D7%99%D7%9D" title="חוק המספרים הגדולים – Hebrew" lang="he" hreflang="he" data-title="חוק המספרים הגדולים" 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/Lielo_skait%C4%BCu_likums" title="Lielo skaitļu likums – Latvian" lang="lv" hreflang="lv" data-title="Lielo skaitļu likums" 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/A_nagy_sz%C3%A1mok_t%C3%B6rv%C3%A9nye" title="A nagy számok törvénye – Hungarian" lang="hu" hreflang="hu" data-title="A nagy számok törvénye" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Li%C4%A1i_tan-numri_kbar" title="Liġi tan-numri kbar – Maltese" lang="mt" hreflang="mt" data-title="Liġi tan-numri kbar" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%98%D1%85_%D1%82%D0%BE%D0%BE%D0%BD%D1%8B_%D1%85%D1%83%D1%83%D0%BB%D1%8C" title="Их тооны хууль – Mongolian" lang="mn" hreflang="mn" data-title="Их тооны хууль" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Wetten_van_de_grote_aantallen" title="Wetten van de grote aantallen – Dutch" lang="nl" hreflang="nl" data-title="Wetten van de grote aantallen" 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/%E5%A4%A7%E6%95%B0%E3%81%AE%E6%B3%95%E5%89%87" title="大数の法則 – Japanese" lang="ja" hreflang="ja" data-title="大数の法則" 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/Store_talls_lov" title="Store talls lov – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Store talls lov" 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-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Lova_om_store_tal" title="Lova om store tal – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Lova om store tal" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/L%C3%A8i_dei_grands_nombres" title="Lèi dei grands nombres – Occitan" lang="oc" hreflang="oc" data-title="Lèi dei grands nombres" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link 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href="/wiki/Law_of_truly_large_numbers" title="Law of truly large numbers">Law of truly large numbers</a>.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Averages of repeated trials converge to the expected value</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist 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a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on <a href="/wiki/Statistics" title="Statistics">statistics</a></td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:Standard_deviation_diagram_micro.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/250px-Standard_deviation_diagram_micro.svg.png" decoding="async" width="250" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/375px-Standard_deviation_diagram_micro.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Standard_deviation_diagram_micro.svg/500px-Standard_deviation_diagram_micro.svg.png 2x" data-file-width="400" data-file-height="200" /></a></span></td></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Probability" title="Probability">Probability</a> <ul><li><a href="/wiki/Probability_axioms" title="Probability axioms">Axioms</a></li></ul></li> <li><a href="/wiki/Determinism" title="Determinism">Determinism</a> <ul><li><a href="/wiki/Deterministic_system" title="Deterministic system">System</a></li></ul></li> <li><a href="/wiki/Indeterminism" title="Indeterminism">Indeterminism</a></li> <li><a href="/wiki/Randomness" title="Randomness">Randomness</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Probability_space" title="Probability space">Probability space</a></li> <li><a href="/wiki/Sample_space" title="Sample space">Sample space</a></li> <li><a href="/wiki/Event_(probability_theory)" title="Event (probability theory)">Event</a> <ul><li><a href="/wiki/Collectively_exhaustive_events" title="Collectively exhaustive events">Collectively exhaustive events</a></li> <li><a href="/wiki/Elementary_event" title="Elementary event">Elementary event</a></li> <li><a href="/wiki/Mutual_exclusivity" title="Mutual exclusivity">Mutual exclusivity</a></li> <li><a href="/wiki/Outcome_(probability)" title="Outcome (probability)">Outcome</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li></ul></li> <li><a href="/wiki/Experiment_(probability_theory)" title="Experiment (probability theory)">Experiment</a> <ul><li><a href="/wiki/Bernoulli_trial" title="Bernoulli trial">Bernoulli trial</a></li></ul></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a> <ul><li><a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli distribution</a></li> <li><a href="/wiki/Binomial_distribution" title="Binomial distribution">Binomial distribution</a></li> <li><a href="/wiki/Exponential_distribution" title="Exponential distribution">Exponential distribution</a></li> <li><a href="/wiki/Normal_distribution" title="Normal distribution">Normal distribution</a></li> <li><a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto distribution</a></li> <li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a></li></ul></li> <li><a href="/wiki/Probability_measure" title="Probability measure">Probability measure</a></li> <li><a href="/wiki/Random_variable" title="Random variable">Random variable</a> <ul><li><a href="/wiki/Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li> <li><a href="/wiki/Continuous_or_discrete_variable" title="Continuous or discrete variable">Continuous or discrete</a></li> <li><a href="/wiki/Expected_value" title="Expected value">Expected value</a></li> <li><a href="/wiki/Variance" title="Variance">Variance</a></li> <li><a href="/wiki/Markov_chain" title="Markov chain">Markov chain</a></li> <li><a href="/wiki/Realization_(probability)" title="Realization (probability)">Observed value</a></li> <li><a href="/wiki/Random_walk" title="Random walk">Random walk</a></li> <li><a href="/wiki/Stochastic_process" title="Stochastic process">Stochastic process</a></li></ul></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Complementary_event" title="Complementary event">Complementary event</a></li> <li><a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">Joint probability</a></li> <li><a href="/wiki/Marginal_distribution" title="Marginal distribution">Marginal probability</a></li> <li><a href="/wiki/Conditional_probability" title="Conditional probability">Conditional probability</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">Independence</a></li> <li><a href="/wiki/Conditional_independence" title="Conditional independence">Conditional independence</a></li> <li><a href="/wiki/Law_of_total_probability" title="Law of total probability">Law of total probability</a></li> <li><a class="mw-selflink selflink">Law of large numbers</a></li> <li><a href="/wiki/Bayes%27_theorem" title="Bayes' theorem">Bayes' theorem</a></li> <li><a href="/wiki/Boole%27s_inequality" title="Boole's inequality">Boole's inequality</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li> <li><a href="/wiki/Tree_diagram_(probability_theory)" title="Tree diagram (probability theory)">Tree diagram</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Probability_fundamentals" title="Template:Probability fundamentals"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Probability_fundamentals" title="Template talk:Probability fundamentals"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Probability_fundamentals" title="Special:EditPage/Template:Probability fundamentals"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Lawoflargenumbers.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Lawoflargenumbers.svg/286px-Lawoflargenumbers.svg.png" decoding="async" width="286" height="191" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Lawoflargenumbers.svg/429px-Lawoflargenumbers.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Lawoflargenumbers.svg/572px-Lawoflargenumbers.svg.png 2x" data-file-width="540" data-file-height="360" /></a><figcaption>An <a href="/wiki/Illustration" title="Illustration">illustration</a> of the law of large numbers using a particular run of rolls of a single <a href="/wiki/Dice" title="Dice">die</a>. As the number of rolls in this run increases, the average of the values of all the results approaches 3.5. Although each run would show a distinctive shape over a small number of throws (at the left), over a large number of rolls (to the right) the shapes would be extremely similar.</figcaption></figure> <p>In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, the <b>law of large numbers</b> (<b>LLN</b>) is a <a href="/wiki/Law_(mathematics)" title="Law (mathematics)">mathematical law</a> that states that the <a href="/wiki/Average" title="Average">average</a> of the results obtained from a large number of independent random samples converges to the true value, if it exists.<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> More formally, the LLN states that given a sample of independent and identically distributed values, the <a href="/wiki/Sample_mean_and_covariance" title="Sample mean and covariance">sample mean</a> converges to the true <a href="/wiki/Mean" title="Mean">mean</a>. </p><p>The LLN is important because it guarantees stable long-term results for the averages of some <a href="/wiki/Randomness" title="Randomness">random</a> <a href="/wiki/Event_(probability_theory)" title="Event (probability theory)">events</a>.<sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><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> For example, while a <a href="/wiki/Casino" title="Casino">casino</a> may lose <a href="/wiki/Money" title="Money">money</a> in a single spin of the <a href="/wiki/Roulette" title="Roulette">roulette</a> wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. Importantly, the law applies (as the name indicates) only when a <i>large number</i> of observations are considered. There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be "balanced" by the others (see the <a href="/wiki/Gambler%27s_fallacy" title="Gambler's fallacy">gambler's fallacy</a>). </p><p>The LLN only applies to the <i>average</i> of the results obtained from repeated trials and claims that this average converges to the expected value; it does not claim that the <i>sum</i> of <i>n</i> results gets close to the expected value times <i>n</i> as <i>n</i> increases. </p><p>Throughout its history, many mathematicians have refined this law. Today, the LLN is used in many fields including statistics, probability theory, economics, and insurance.<sup id="cite_ref-:1_3-0" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Law_of_large_numbers&action=edit&section=1" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For example, a single roll of a fair, six-sided die produces one of the numbers 1, 2, 3, 4, 5, or 6, each with equal <a href="/wiki/Probability" title="Probability">probability</a>. Therefore, the <a href="/wiki/Expected_value" title="Expected value">expected value</a> of the average of the rolls is: </p><p><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+2+3+4+5+6}{6}}=3.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>4</mn> <mo>+</mo> <mn>5</mn> <mo>+</mo> <mn>6</mn> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>=</mo> <mn>3.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1+2+3+4+5+6}{6}}=3.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d28b04241bda3525a74f5583534c3321047098c7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.083ex; height:5.176ex;" alt="{\displaystyle {\frac {1+2+3+4+5+6}{6}}=3.5}"></span> </p><p>According to the law of large numbers, if a large number of six-sided dice are rolled, the average of their values (sometimes called the <a href="/wiki/Sample_mean" class="mw-redirect" title="Sample mean">sample mean</a>) will approach 3.5, with the precision increasing as more dice are rolled. </p><p>It follows from the law of large numbers that the <a href="/wiki/Empirical_probability" title="Empirical probability">empirical probability</a> of success in a series of <a href="/wiki/Bernoulli_trial" title="Bernoulli trial">Bernoulli trials</a> will converge to the theoretical probability. For a <a href="/wiki/Bernoulli_random_variable" class="mw-redirect" title="Bernoulli random variable">Bernoulli random variable</a>, the expected value is the theoretical probability of success, and the average of <i>n</i> such variables (assuming they are <a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">independent and identically distributed (i.i.d.)</a>) is precisely the relative frequency. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Law_of_large_numbers_(black_%26_red_balls).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Law_of_large_numbers_%28black_%26_red_balls%29.png/295px-Law_of_large_numbers_%28black_%26_red_balls%29.png" decoding="async" width="295" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Law_of_large_numbers_%28black_%26_red_balls%29.png/443px-Law_of_large_numbers_%28black_%26_red_balls%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Law_of_large_numbers_%28black_%26_red_balls%29.png/590px-Law_of_large_numbers_%28black_%26_red_balls%29.png 2x" data-file-width="1800" data-file-height="600" /></a><figcaption> This image illustrates the convergence of relative frequencies to their theoretical probabilities. The probability of picking a red ball from a sack is 0.4 and black ball is 0.6. The left plot shows the relative frequency of picking a black ball, and the right plot shows the relative frequency of picking a red ball, both over 10,000 trials. As the number of trials increases, the relative frequencies approach their respective theoretical probabilities, demonstrating the Law of Large Numbers.</figcaption></figure> <p>For example, a <a href="/wiki/Fair_coin" title="Fair coin">fair coin</a> toss is a Bernoulli trial. When a fair coin is flipped once, the theoretical probability that the outcome will be heads is equal to <style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.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="frac"><span class="num">1</span>⁄<span class="den">2</span></span>. Therefore, according to the law of large numbers, the proportion of heads in a "large" number of coin flips "should be" roughly <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>. In particular, the proportion of heads after <i>n</i> flips will <a href="/wiki/Almost_surely" title="Almost surely">almost surely</a> <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">converge</a> to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span> as <i>n</i> approaches infinity. </p><p>Although the proportion of heads (and tails) approaches <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>, almost surely the <a href="/wiki/Absolute_difference" title="Absolute difference">absolute difference</a> in the number of heads and tails will become large as the number of flips becomes large. That is, the probability that the absolute difference is a small number approaches zero as the number of flips becomes large. Also, almost surely the ratio of the absolute difference to the number of flips will approach zero. Intuitively, the expected difference grows, but at a slower rate than the number of flips. </p><p>Another good example of the LLN is the <a href="/wiki/Monte_Carlo_method" title="Monte Carlo method">Monte Carlo method</a>. These methods are a broad class of <a href="/wiki/Computation" title="Computation">computational</a> <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> that rely on repeated <a href="/wiki/Random_sampling" class="mw-redirect" title="Random sampling">random sampling</a> to obtain numerical results. The larger the number of repetitions, the better the approximation tends to be. The reason that this method is important is mainly that, sometimes, it is difficult or impossible to use other approaches.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Limitation">Limitation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Law_of_large_numbers&action=edit&section=2" title="Edit section: Limitation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The average of the results obtained from a large number of trials may fail to converge in some cases. For instance, the average of <i>n</i> results taken from the <a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy distribution</a> or some <a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto distributions</a> (α<1) will not converge as <i>n</i> becomes larger; the reason is <a href="/wiki/Heavy-tailed_distribution" title="Heavy-tailed distribution">heavy tails</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The Cauchy distribution and the Pareto distribution represent two cases: the Cauchy distribution does not have an expectation,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> whereas the expectation of the Pareto distribution (<i>α</i><1) is infinite.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> One way to generate the Cauchy-distributed example is where the random numbers equal the <a href="/wiki/Tangent" title="Tangent">tangent</a> of an angle uniformly distributed between −90° and +90°.<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> The <a href="/wiki/Median" title="Median">median</a> is zero, but the expected value does not exist, and indeed the average of <i>n</i> such variables have the same distribution as one such variable. It does not converge in probability toward zero (or any other value) as <i>n</i> goes to infinity. </p><p>And if the trials embed a <a href="/wiki/Selection_bias" title="Selection bias">selection bias</a>, typical in human economic/rational behaviour, the law of large numbers does not help in solving the bias. Even if the number of trials is increased the selection bias remains. </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=Law_of_large_numbers&action=edit&section=3" title="Edit section: History"><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:DiffusionMicroMacro.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/DiffusionMicroMacro.gif/250px-DiffusionMicroMacro.gif" decoding="async" width="250" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/4/4d/DiffusionMicroMacro.gif 1.5x" data-file-width="360" data-file-height="300" /></a><figcaption><a href="/wiki/Molecular_diffusion" title="Molecular diffusion">Diffusion</a> is an example of the law of large numbers. Initially, there are <a href="/wiki/Solute" class="mw-redirect" title="Solute">solute</a> molecules on the left side of a barrier (magenta line) and none on the right. The barrier is removed, and the solute diffuses to fill the whole container.<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="plainlist" style="margin-top:1em"><ul><li><i>Top:</i> With a single molecule, the motion appears to be quite random.</li><li><i>Middle:</i> With more molecules, there is clearly a trend where the solute fills the container more and more uniformly, but there are also random fluctuations.</li><li><i>Bottom:</i> With an enormous number of solute molecules (too many to see), the randomness is essentially gone: The solute appears to move smoothly and systematically from high-concentration areas to low-concentration areas. In realistic situations, chemists can describe diffusion as a deterministic macroscopic phenomenon (see <a href="/wiki/Fick%27s_law" class="mw-redirect" title="Fick's law">Fick's laws</a>), despite its underlying random nature.</li></ul></div></figcaption></figure> <p>The Italian mathematician <a href="/wiki/Gerolamo_Cardano" title="Gerolamo Cardano">Gerolamo Cardano</a> (1501–1576) stated without proof that the accuracies of empirical statistics tend to improve with the number of trials.<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><sup id="cite_ref-:1_3-1" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> This was then formalized as a law of large numbers. A special form of the LLN (for a binary random variable) was first proved by <a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a>.<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><sup id="cite_ref-:1_3-2" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> It took him over 20 years to develop a sufficiently rigorous mathematical proof which was published in his <span title="Latin-language text"><i lang="la"><a href="/wiki/Ars_Conjectandi" title="Ars Conjectandi">Ars Conjectandi</a></i></span> (<i>The Art of Conjecturing</i>) in 1713. He named this his "Golden Theorem" but it became generally known as "<b>Bernoulli's theorem</b>". This should not be confused with <a href="/wiki/Bernoulli%27s_principle" title="Bernoulli's principle">Bernoulli's principle</a>, named after Jacob Bernoulli's nephew <a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a>. In 1837, <a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">S. D. Poisson</a> further described it under the name <span title="French-language text"><i lang="fr">"la loi des grands nombres"</i></span> ("the law of large numbers").<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><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><sup id="cite_ref-:1_3-3" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Thereafter, it was known under both names, but the "law of large numbers" is most frequently used. </p><p>After Bernoulli and Poisson published their efforts, other mathematicians also contributed to refinement of the law, including <a href="/wiki/Pafnuty_Chebyshev" title="Pafnuty Chebyshev">Chebyshev</a>,<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> <a href="/wiki/Andrey_Markov" title="Andrey Markov">Markov</a>, <a href="/wiki/%C3%89mile_Borel" title="Émile Borel">Borel</a>, <a href="/wiki/Francesco_Paolo_Cantelli" title="Francesco Paolo Cantelli">Cantelli</a>, <a href="/wiki/Andrey_Kolmogorov" title="Andrey Kolmogorov">Kolmogorov</a> and <a href="/wiki/Aleksandr_Khinchin" title="Aleksandr Khinchin">Khinchin</a>.<sup id="cite_ref-:1_3-4" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Markov showed that the law can apply to a random variable that does not have a finite variance under some other weaker assumption, and Khinchin showed in 1929 that if the series consists of independent identically distributed random variables, it suffices that the <a href="/wiki/Expected_value" title="Expected value">expected value</a> exists for the weak law of large numbers to be true.<sup id="cite_ref-FOOTNOTESeneta2013_14-0" class="reference"><a href="#cite_note-FOOTNOTESeneta2013-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-EncMath_15-0" class="reference"><a href="#cite_note-EncMath-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> These further studies have given rise to two prominent forms of the LLN. One is called the "weak" law and the other the "strong" law, in reference to two different modes of <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">convergence</a> of the cumulative sample means to the expected value; in particular, as explained below, the strong form implies the weak.<sup id="cite_ref-FOOTNOTESeneta2013_14-1" class="reference"><a href="#cite_note-FOOTNOTESeneta2013-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Forms">Forms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Law_of_large_numbers&action=edit&section=4" title="Edit section: Forms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are two different versions of the <b>law of large numbers</b> that are described below. They are called the<i> <b>strong law</b> of large numbers</i> and the <i><b>weak law</b> of large numbers</i>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_1-2" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Stated for the case where <i>X</i><sub>1</sub>, <i>X</i><sub>2</sub>, ... is an infinite sequence of <a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">independent and identically distributed (i.i.d.)</a> <a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integrable</a> random variables with expected value E(<i>X</i><sub>1</sub>) = E(<i>X</i><sub>2</sub>) = ... = <i>μ</i>, both versions of the law state that the sample average </p><p><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 {\overline {X}}_{n}={\frac {1}{n}}(X_{1}+\cdots +X_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <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> <mo 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 stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}_{n}={\frac {1}{n}}(X_{1}+\cdots +X_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3557f2a15de9f9ce914f8ef0b013a3caa5d77b0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.014ex; height:5.176ex;" alt="{\displaystyle {\overline {X}}_{n}={\frac {1}{n}}(X_{1}+\cdots +X_{n})}"></span> </p><p>converges to the expected value: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 {\overline {X}}_{n}\to \mu \quad {\textrm {as}}\ n\to \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mi>μ</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>as</mtext> </mrow> </mrow> <mtext> </mtext> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}_{n}\to \mu \quad {\textrm {as}}\ n\to \infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d125a1d2c089f1810b2c8e06c29ea8358a3db88" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.327ex; height:3.509ex;" alt="{\displaystyle {\overline {X}}_{n}\to \mu \quad {\textrm {as}}\ n\to \infty .}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span>)</b></td></tr></tbody></table> <p>(Lebesgue integrability of <i>X<sub>j</sub></i> means that the expected value E(<i>X<sub>j</sub></i>) exists according to Lebesgue integration and is finite. It does <i>not</i> mean that the associated probability measure is <a href="/wiki/Absolutely_continuous" class="mw-redirect" title="Absolutely continuous">absolutely continuous</a> with respect to <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a>.) </p><p>Introductory probability texts often additionally assume identical finite <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 \operatorname {Var} (X_{i})=\sigma ^{2}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} (X_{i})=\sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd1be36beafcc2454eaa20fb8e09a00cc2e2fa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.834ex; height:3.176ex;" alt="{\displaystyle \operatorname {Var} (X_{i})=\sigma ^{2}}"></span> (for all <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>) and no correlation between random variables. In that case, the variance of the average of n random variables is </p><p><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 \operatorname {Var} ({\overline {X}}_{n})=\operatorname {Var} ({\tfrac {1}{n}}(X_{1}+\cdots +X_{n}))={\frac {1}{n^{2}}}\operatorname {Var} (X_{1}+\cdots +X_{n})={\frac {n\sigma ^{2}}{n^{2}}}={\frac {\sigma ^{2}}{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo 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 stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>Var</mi> <mo>⁡</mo> <mo 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 stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} ({\overline {X}}_{n})=\operatorname {Var} ({\tfrac {1}{n}}(X_{1}+\cdots +X_{n}))={\frac {1}{n^{2}}}\operatorname {Var} (X_{1}+\cdots +X_{n})={\frac {n\sigma ^{2}}{n^{2}}}={\frac {\sigma ^{2}}{n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3afd10e29bec1e03a91c726c4ddbfd29ccdc40e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:77.461ex; height:6.009ex;" alt="{\displaystyle \operatorname {Var} ({\overline {X}}_{n})=\operatorname {Var} ({\tfrac {1}{n}}(X_{1}+\cdots +X_{n}))={\frac {1}{n^{2}}}\operatorname {Var} (X_{1}+\cdots +X_{n})={\frac {n\sigma ^{2}}{n^{2}}}={\frac {\sigma ^{2}}{n}}.}"></span> </p><p>which can be used to shorten and simplify the proofs. This assumption of finite <a href="/wiki/Variance" title="Variance">variance</a> is <i>not necessary</i>. Large or infinite variance will make the convergence slower, but the LLN holds anyway.<sup id="cite_ref-TaoBlog_17-0" class="reference"><a href="#cite_note-TaoBlog-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Independence_(probability_theory)#More_than_two_random_variables" title="Independence (probability theory)">Mutual independence</a> of the random variables can be replaced by <a href="/wiki/Pairwise_independence" title="Pairwise independence">pairwise independence</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> or <a href="/wiki/Exchangeable_random_variables" title="Exchangeable random variables">exchangeability</a><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> in both versions of the law. </p><p>The difference between the strong and the weak version is concerned with the mode of convergence being asserted. For interpretation of these modes, see <a href="/wiki/Convergence_of_random_variables" title="Convergence of random variables">Convergence of random variables</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Weak_law">Weak law</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Law_of_large_numbers&action=edit&section=5" title="Edit section: Weak law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 tright"><div class="thumbinner multiimageinner" style="width:212px;max-width:212px"><div class="trow"><div class="tsingle" style="width:52px;max-width:52px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Blank300.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Blank300.png/50px-Blank300.png" decoding="async" width="50" height="0" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Blank300.png/75px-Blank300.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/Blank300.png/100px-Blank300.png 2x" data-file-width="300" data-file-height="1" /></a></span></div></div><div class="tsingle" style="width:102px;max-width:102px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Lawoflargenumbersanimation2.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/4/49/Lawoflargenumbersanimation2.gif" decoding="async" width="100" height="169" class="mw-file-element" data-file-width="100" data-file-height="169" /></a></span></div></div><div class="tsingle" style="width:52px;max-width:52px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Blank300.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Blank300.png/50px-Blank300.png" decoding="async" width="50" height="0" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Blank300.png/75px-Blank300.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/Blank300.png/100px-Blank300.png 2x" data-file-width="300" data-file-height="1" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Simulation illustrating the law of large numbers. Each frame, a coin that is red on one side and blue on the other is flipped, and a dot is added in the corresponding column. A pie chart shows the proportion of red and blue so far. Notice that while the proportion varies significantly at first, it approaches 50% as the number of trials increases.</div></div></div></div> <p>The <b>weak law of large numbers</b> (also called <a href="/wiki/Aleksandr_Khinchin" title="Aleksandr Khinchin">Khinchin</a>'s law) states that given a collection of <a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">independent and identically distributed</a> (iid) samples from a random variable with finite mean, the sample mean <a href="/wiki/Convergence_in_probability" class="mw-redirect" title="Convergence in probability">converges in probability</a> to the expected value<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> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 {\overline {X}}_{n}\ {\overset {P}{\rightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo stretchy="false">→</mo> <mi>P</mi> </mover> </mrow> <mtext> </mtext> <mi>μ</mi> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>when</mtext> </mrow> </mrow> <mtext> </mtext> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}_{n}\ {\overset {P}{\rightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e9cd779400133a1b72dc4ac5695c46b44f1740f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.737ex; height:4.176ex;" alt="{\displaystyle {\overline {X}}_{n}\ {\overset {P}{\rightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span>)</b></td></tr></tbody></table> <p>That is, for any positive number <i>ε</i>, </p><p><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 }\Pr \!\left(\,|{\overline {X}}_{n}-\mu |<\varepsilon \,\right)=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> <mo movablelimits="true" form="prefix">Pr</mo> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> <mspace width="thinmathspace" /> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }\Pr \!\left(\,|{\overline {X}}_{n}-\mu |<\varepsilon \,\right)=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa7282cd85c15acdfb0372de2ab4eddd9123214d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.679ex; height:4.843ex;" alt="{\displaystyle \lim _{n\to \infty }\Pr \!\left(\,|{\overline {X}}_{n}-\mu |<\varepsilon \,\right)=1.}"></span> </p><p>Interpreting this result, the weak law states that for any nonzero margin specified (<i>ε</i>), no matter how small, with a sufficiently large sample there will be a very high probability that the average of the observations will be close to the expected value; that is, within the margin. </p><p>As mentioned earlier, the weak law applies in the case of i.i.d. random variables, but it also applies in some other cases. For example, the variance may be different for each random variable in the series, keeping the expected value constant. If the variances are bounded, then the law applies, as shown by <a href="/wiki/Pafnuty_Chebyshev" title="Pafnuty Chebyshev">Chebyshev</a> as early as 1867. (If the expected values change during the series, then we can simply apply the law to the average deviation from the respective expected values. The law then states that this converges in probability to zero.) In fact, Chebyshev's proof works so long as the variance of the average of the first <i>n</i> values goes to zero as <i>n</i> goes to infinity.<sup id="cite_ref-EncMath_15-1" class="reference"><a href="#cite_note-EncMath-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> As an example, assume that each random variable in the series follows a <a href="/wiki/Gaussian_distribution" class="mw-redirect" title="Gaussian distribution">Gaussian distribution</a> (normal distribution) with mean zero, but with variance equal 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 2n/\log(n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2n/\log(n+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7567503dc3082397f70bb7a8a9418dbe0bfd6e67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.285ex; height:2.843ex;" alt="{\displaystyle 2n/\log(n+1)}"></span>, which is not bounded. At each stage, the average will be normally distributed (as the average of a set of normally distributed variables). The variance of the sum is equal to the sum of the variances, which is <a href="/wiki/Asymptotic" class="mw-redirect" title="Asymptotic">asymptotic</a> 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 n^{2}/\log n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}/\log n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30ac16b39b51cc8b62dc92cb68ffa460d7ed117b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.752ex; height:3.176ex;" alt="{\displaystyle n^{2}/\log n}"></span>. The variance of the average is therefore asymptotic 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 1/\log n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/\log n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcaa2f4e8b4f2c5839cf4e35d733f3db591f83dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.466ex; height:2.843ex;" alt="{\displaystyle 1/\log n}"></span> and goes to zero. </p><p>There are also examples of the weak law applying even though the expected value does not exist. </p> <div class="mw-heading mw-heading3"><h3 id="Strong_law">Strong law</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Law_of_large_numbers&action=edit&section=6" title="Edit section: Strong law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>strong law of large numbers</b> (also called <a href="/wiki/Andrey_Kolmogorov" title="Andrey Kolmogorov">Kolmogorov</a>'s law) states that the sample average <a href="/wiki/Almost_sure_convergence" class="mw-redirect" title="Almost sure convergence">converges almost surely</a> to the expected value<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 {\overline {X}}_{n}\ {\overset {\text{a.s.}}{\longrightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo stretchy="false">⟶</mo> <mtext>a.s.</mtext> </mover> </mrow> <mtext> </mtext> <mi>μ</mi> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>when</mtext> </mrow> </mrow> <mtext> </mtext> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}_{n}\ {\overset {\text{a.s.}}{\longrightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d48d6e215ec6f6925f629a19acafbe91cf2273" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.219ex; height:3.676ex;" alt="{\displaystyle {\overline {X}}_{n}\ {\overset {\text{a.s.}}{\longrightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span>)</b></td></tr></tbody></table> <p>That is, </p><p><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 \Pr \!\left(\lim _{n\to \infty }{\overline {X}}_{n}=\mu \right)=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow> <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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>μ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr \!\left(\lim _{n\to \infty }{\overline {X}}_{n}=\mu \right)=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/befeda3c4b77efb2cf7835a9569edaadebd978e9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.687ex; height:4.843ex;" alt="{\displaystyle \Pr \!\left(\lim _{n\to \infty }{\overline {X}}_{n}=\mu \right)=1.}"></span> </p><p>What this means is that the probability that, as the number of trials <i>n</i> goes to infinity, the average of the observations converges to the expected value, is equal to one. The modern proof of the strong law is more complex than that of the weak law, and relies on passing to an appropriate subsequence.<sup id="cite_ref-TaoBlog_17-1" class="reference"><a href="#cite_note-TaoBlog-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>The strong law of large numbers can itself be seen as a special case of the <a href="/wiki/Ergodic_theory#Ergodic_theorems" title="Ergodic theory">pointwise ergodic theorem</a>. This view justifies the intuitive interpretation of the expected value (for Lebesgue integration only) of a random variable when sampled repeatedly as the "long-term average". </p><p>Law 3 is called the strong law because random variables which converge strongly (almost surely) are guaranteed to converge weakly (in probability). However the weak law is known to hold in certain conditions where the strong law does not hold and then the convergence is only weak (in probability). See <a href="#Differences_between_the_weak_law_and_the_strong_law">differences between the weak law and the strong law</a>. </p><p>The strong law applies to independent identically distributed random variables having an expected value (like the weak law). This was proved by Kolmogorov in 1930. It can also apply in other cases. Kolmogorov also showed, in 1933, that if the variables are independent and identically distributed, then for the average to converge almost surely on <i>something</i> (this can be considered another statement of the strong law), it is necessary that they have an expected value (and then of course the average will converge almost surely on that).<sup id="cite_ref-EMStrong_22-0" class="reference"><a href="#cite_note-EMStrong-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p>If the summands are independent but not identically distributed, then </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 {\overline {X}}_{n}-\operatorname {E} {\big [}{\overline {X}}_{n}{\big ]}\ {\overset {\text{a.s.}}{\longrightarrow }}\ 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo stretchy="false">⟶</mo> <mtext>a.s.</mtext> </mover> </mrow> <mtext> </mtext> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}_{n}-\operatorname {E} {\big [}{\overline {X}}_{n}{\big ]}\ {\overset {\text{a.s.}}{\longrightarrow }}\ 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d78fdcd0502a7beb22812467a91e5e8eb6acc95" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.225ex; height:3.843ex;" alt="{\displaystyle {\overline {X}}_{n}-\operatorname {E} {\big [}{\overline {X}}_{n}{\big ]}\ {\overset {\text{a.s.}}{\longrightarrow }}\ 0,}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span>)</b></td></tr></tbody></table> <p>provided that each <i>X</i><sub><i>k</i></sub> has a finite second moment and </p><p><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 _{k=1}^{\infty }{\frac {1}{k^{2}}}\operatorname {Var} [X_{k}]<\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k^{2}}}\operatorname {Var} [X_{k}]<\infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/849940f493b14c4f019c225b494d3afcc1b6ca29" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.424ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k^{2}}}\operatorname {Var} [X_{k}]<\infty .}"></span> </p><p>This statement is known as <i>Kolmogorov's strong law</i>, see e.g. <a href="#CITEREFSenSinger1993">Sen & Singer (1993</a>, Theorem 2.3.10). </p> <div class="mw-heading mw-heading3"><h3 id="Differences_between_the_weak_law_and_the_strong_law">Differences between the weak law and the strong law</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Law_of_large_numbers&action=edit&section=7" title="Edit section: Differences between the weak law and the strong law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>weak law</i> states that for a specified large <i>n</i>, the 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 {\overline {X}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55a63d07fe7c766a92154413faf8456d31c29fca" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.671ex; width:3.35ex; height:3.343ex;" aria-hidden="true" alt="{\displaystyle {\overline {X}}_{n}}"></span> is likely to be near <i>μ</i>.<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> Thus, it leaves open the possibility 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="{\displaystyle |{\overline {X}}_{n}-\mu |>\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\overline {X}}_{n}-\mu |>\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87264fcd956c78c10dfd2961aacdb7d876ec7a2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:13.067ex; height:3.509ex;" aria-hidden="true" alt="{\displaystyle |{\overline {X}}_{n}-\mu |>\varepsilon }"></span> happens an infinite number of times, although at infrequent intervals. (Not necessarily <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 |{\overline {X}}_{n}-\mu |\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\overline {X}}_{n}-\mu |\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d16455f9fd64f0f4ea4ec2a4d58b3ff171276d4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:13.146ex; height:3.509ex;" aria-hidden="true" alt="{\displaystyle |{\overline {X}}_{n}-\mu |\neq 0}"></span> for all <i>n</i>). </p><p>The <i>strong law</i> shows that this <a href="/wiki/Almost_surely" title="Almost surely">almost surely</a> will not occur. It does not imply that with probability 1, we have that for any <span class="texhtml"><i>ε</i> > 0</span> the inequality <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 |{\overline {X}}_{n}-\mu |<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\overline {X}}_{n}-\mu |<\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51afeb452715e1bbc148a8e223062caf8d7071c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.838ex; width:13.067ex; height:3.509ex;" aria-hidden="true" alt="{\displaystyle |{\overline {X}}_{n}-\mu |<\varepsilon }"></span> holds for all large enough <i>n</i>, since the convergence is not necessarily uniform on the set where it holds.<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> </p><p>The strong law does not hold in the following cases, but the weak law does.<sup id="cite_ref-Weak_law_converges_to_constant_25-0" class="reference"><a href="#cite_note-Weak_law_converges_to_constant-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><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> </p> <div><ol><li>Let X be an <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponentially</a> distributed random variable with parameter 1. The random variable <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 \sin(X)e^{X}X^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(X)e^{X}X^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e299f779628fd1a8d75a29013da7e5c24098a411" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.69ex; height:3.176ex;" alt="{\displaystyle \sin(X)e^{X}X^{-1}}"></span> has no expected value according to Lebesgue integration, but using conditional convergence and interpreting the integral as a <a href="/wiki/Dirichlet_integral" title="Dirichlet integral">Dirichlet integral</a>, which is an improper <a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a>, we can say: <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 E\left({\frac {\sin(X)e^{X}}{X}}\right)=\ \int _{x=0}^{\infty }{\frac {\sin(x)e^{x}}{x}}e^{-x}dx={\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> </mrow> <mi>X</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mtext> </mtext> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mrow> <mi>x</mi> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> </mrow> </msup> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left({\frac {\sin(X)e^{X}}{X}}\right)=\ \int _{x=0}^{\infty }{\frac {\sin(x)e^{x}}{x}}e^{-x}dx={\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a2c8eaddf7bbc8ce9a71ea8bd5b1bdb7a54e4b3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.493ex; height:7.509ex;" alt="{\displaystyle E\left({\frac {\sin(X)e^{X}}{X}}\right)=\ \int _{x=0}^{\infty }{\frac {\sin(x)e^{x}}{x}}e^{-x}dx={\frac {\pi }{2}}}"></span></li><li>Let X be a <a href="/wiki/Geometric_distribution" title="Geometric distribution">geometrically</a> distributed random variable with probability 0.5. The random variable <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 2^{X}(-1)^{X}X^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{X}(-1)^{X}X^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c038343af9b9e3a6e4fef27d17e9fe34f331bbf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.537ex; height:3.176ex;" alt="{\displaystyle 2^{X}(-1)^{X}X^{-1}}"></span> does not have an expected value in the conventional sense because the infinite <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a> is not absolutely convergent, but using conditional convergence, we can say: <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 E\left({\frac {2^{X}(-1)^{X}}{X}}\right)=\ \sum _{x=1}^{\infty }{\frac {2^{x}(-1)^{x}}{x}}2^{-x}=-\ln(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> </mrow> <mi>X</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mtext> </mtext> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mrow> <mi>x</mi> </mfrac> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left({\frac {2^{X}(-1)^{X}}{X}}\right)=\ \sum _{x=1}^{\infty }{\frac {2^{x}(-1)^{x}}{x}}2^{-x}=-\ln(2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfebf3b91b31b4bd9f4890870b4d6000e8975a64" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.637ex; height:7.509ex;" alt="{\displaystyle E\left({\frac {2^{X}(-1)^{X}}{X}}\right)=\ \sum _{x=1}^{\infty }{\frac {2^{x}(-1)^{x}}{x}}2^{-x}=-\ln(2)}"></span></li><li>If the <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> of a random variable 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 {\begin{cases}1-F(x)&={\frac {e}{2x\ln(x)}},&x\geq e\\F(x)&={\frac {e}{-2x\ln(-x)}},&x\leq -e\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>e</mi> <mrow> <mn>2</mn> <mi>x</mi> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mi>x</mi> <mo>≥</mo> <mi>e</mi> </mtd> </mtr> <mtr> <mtd> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>e</mi> <mrow> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mi>x</mi> <mo>≤</mo> <mo>−</mo> <mi>e</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}1-F(x)&={\frac {e}{2x\ln(x)}},&x\geq e\\F(x)&={\frac {e}{-2x\ln(-x)}},&x\leq -e\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/517c633d41244b5c3a50245f09aa30d7a9d45c6b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:35.706ex; height:7.843ex;" alt="{\displaystyle {\begin{cases}1-F(x)&={\frac {e}{2x\ln(x)}},&x\geq e\\F(x)&={\frac {e}{-2x\ln(-x)}},&x\leq -e\end{cases}}}"></span> then it has no expected value, but the weak law is true.<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><sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup></li><li>Let <i>X</i><sub><i>k</i></sub> be plus or minus <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 {\sqrt {k/\log \log \log k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>k</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {k/\log \log \log k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61ba30676d65593a89136b99d79d25d10a04252b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.372ex; height:3.343ex;" alt="{\textstyle {\sqrt {k/\log \log \log k}}}"></span> (starting at sufficiently large <i>k</i> so that the denominator is positive) with probability <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span> for each.<sup id="cite_ref-EMStrong_22-1" class="reference"><a href="#cite_note-EMStrong-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> The variance of <i>X</i><sub><i>k</i></sub> is 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 k/\log \log \log k.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>k</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k/\log \log \log k.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd34bd7a050bfee85ed9a2acdeb49ac717f5f503" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.695ex; height:2.843ex;" alt="{\displaystyle k/\log \log \log k.}"></span> Kolmogorov's strong law does not apply because the partial sum in his criterion up to <i>k</i> = <i>n</i> is asymptotic 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 \log n/\log \log \log n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log n/\log \log \log n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddf19b230e29269bc0858e4090d9b114a870ab4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.774ex; height:2.843ex;" alt="{\displaystyle \log n/\log \log \log n}"></span> and this is unbounded. If we replace the random variables with Gaussian variables having the same variances, namely <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 {\sqrt {k/\log \log \log k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>k</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {k/\log \log \log k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61ba30676d65593a89136b99d79d25d10a04252b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.372ex; height:3.343ex;" alt="{\textstyle {\sqrt {k/\log \log \log k}}}"></span>, then the average at any point will also be normally distributed. The width of the distribution of the average will tend toward zero (standard deviation asymptotic 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 1/{\sqrt {2\log \log \log 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> <mn>2</mn> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>log</mi> <mo>⁡</mo> <mi>n</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 1/{\sqrt {2\log \log \log n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cfb16df11a96e70cee926f751edb4574dfcc815" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.669ex; height:3.343ex;" alt="{\textstyle 1/{\sqrt {2\log \log \log n}}}"></span>), but for a given <i>ε</i>, there is probability which does not go to zero with <i>n</i>, while the average sometime after the <i>n</i>th trial will come back up to <i>ε</i>. Since the width of the distribution of the average is not zero, it must have a positive lower bound <i>p</i>(<i>ε</i>), which means there is a probability of at least <i>p</i>(<i>ε</i>) that the average will attain ε after <i>n</i> trials. It will happen with probability <i>p</i>(<i>ε</i>)/2 before some <i>m</i> which depends on <i>n</i>. But even after <i>m</i>, there is still a probability of at least <i>p</i>(<i>ε</i>) that it will happen. (This seems to indicate that <i>p</i>(<i>ε</i>)=1 and the average will attain ε an infinite number of times.)</li></ol></div> <div class="mw-heading mw-heading3"><h3 id="Uniform_laws_of_large_numbers">Uniform laws of large numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Law_of_large_numbers&action=edit&section=8" title="Edit section: Uniform laws of large numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are extensions of the law of large numbers to collections of estimators, where the convergence is uniform over the collection; thus the name <i>uniform law of large numbers</i>. </p><p>Suppose <i>f</i>(<i>x</i>,<i>θ</i>) is some <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> defined for <i>θ</i> ∈ Θ, and continuous in <i>θ</i>. Then for any fixed <i>θ</i>, the sequence {<i>f</i>(<i>X</i><sub>1</sub>,<i>θ</i>), <i>f</i>(<i>X</i><sub>2</sub>,<i>θ</i>), ...} will be a sequence of independent and identically distributed random variables, such that the sample mean of this sequence converges in probability to E[<i>f</i>(<i>X</i>,<i>θ</i>)]. This is the <i>pointwise</i> (in <i>θ</i>) convergence. </p><p>A particular example of a <b>uniform law of large numbers</b> states the conditions under which the convergence happens <i>uniformly</i> in <i>θ</i>. If<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><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> </p> <ol><li><i>Θ</i> is compact,</li> <li><i>f</i>(<i>x</i>,<i>θ</i>) is continuous at each <i>θ</i> ∈ Θ for <a href="/wiki/Almost_everywhere" title="Almost everywhere">almost all</a> <i>x</i>s, and measurable function of <i>x</i> at each <i>θ</i>.</li> <li>there exists a <a href="/wiki/Dominated_convergence_theorem" title="Dominated convergence theorem">dominating</a> function <i>d</i>(<i>x</i>) such that E[<i>d</i>(<i>X</i>)] < ∞, and <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\|f(x,\theta )\right\|\leq d(x)\quad {\text{for all}}\ \theta \in \Theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo symmetric="true">‖</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>≤</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for all</mtext> </mrow> <mtext> </mtext> <mi>θ</mi> <mo>∈</mo> <mi mathvariant="normal">Θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\|f(x,\theta )\right\|\leq d(x)\quad {\text{for all}}\ \theta \in \Theta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/315b2fa353909618ab22a100b078fe99fb608df3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.432ex; height:2.843ex;" alt="{\displaystyle \left\|f(x,\theta )\right\|\leq d(x)\quad {\text{for all}}\ \theta \in \Theta .}"></span></li></ol> <p>Then E[<i>f</i>(<i>X</i>,<i>θ</i>)] is continuous in <i>θ</i>, and </p><p><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 \sup _{\theta \in \Theta }\left\|{\frac {1}{n}}\sum _{i=1}^{n}f(X_{i},\theta )-\operatorname {E} [f(X,\theta )]\right\|{\overset {\mathrm {P} }{\rightarrow }}\ 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> <mo>∈</mo> <mi mathvariant="normal">Θ</mi> </mrow> </munder> <mrow> <mo symmetric="true">‖</mo> <mrow> <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> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> </mrow> </mover> </mrow> <mtext> </mtext> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup _{\theta \in \Theta }\left\|{\frac {1}{n}}\sum _{i=1}^{n}f(X_{i},\theta )-\operatorname {E} [f(X,\theta )]\right\|{\overset {\mathrm {P} }{\rightarrow }}\ 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb54a1fa254d7f7ed846e6cabc7a8a1f789a92aa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.52ex; height:7.176ex;" alt="{\displaystyle \sup _{\theta \in \Theta }\left\|{\frac {1}{n}}\sum _{i=1}^{n}f(X_{i},\theta )-\operatorname {E} [f(X,\theta )]\right\|{\overset {\mathrm {P} }{\rightarrow }}\ 0.}"></span> </p><p>This result is useful to derive consistency of a large class of estimators (see <a href="/wiki/Extremum_estimator" title="Extremum estimator">Extremum estimator</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Borel's_law_of_large_numbers"><span id="Borel.27s_law_of_large_numbers"></span>Borel's law of large numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Law_of_large_numbers&action=edit&section=9" title="Edit section: Borel's law of large numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Borel's law of large numbers</b>, named after <a href="/wiki/%C3%89mile_Borel" title="Émile Borel">Émile Borel</a>, states that if an experiment is repeated a large number of times, independently under identical conditions, then the proportion of times that any specified event is expected to occur approximately equals the probability of the event's occurrence on any particular trial; the larger the number of repetitions, the better the approximation tends to be. More precisely, if <i>E</i> denotes the event in question, <i>p</i> its probability of occurrence, and <i>N<sub>n</sub></i>(<i>E</i>) the number of times <i>E</i> occurs in the first <i>n</i> trials, then with probability one,<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> <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 {N_{n}(E)}{n}}\to p{\text{ as }}n\to \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">→</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> as </mtext> </mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {N_{n}(E)}{n}}\to p{\text{ as }}n\to \infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/554308716ea75ab0bd65661624e0eee87e390705" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.509ex; height:5.676ex;" alt="{\displaystyle {\frac {N_{n}(E)}{n}}\to p{\text{ as }}n\to \infty .}"></span> </p><p>This theorem makes rigorous the intuitive notion of probability as the expected long-run relative frequency of an event's occurrence. It is a special case of any of several more general laws of large numbers in probability theory. </p><p><b><a href="/wiki/Chebyshev%27s_inequality" title="Chebyshev's inequality">Chebyshev's inequality</a></b>. Let <i>X</i> be a <a href="/wiki/Random_variable" title="Random variable">random variable</a> with finite <a href="/wiki/Expected_value" title="Expected value">expected value</a> <i>μ</i> and finite non-zero <a href="/wiki/Variance" title="Variance">variance</a> <i>σ</i><sup>2</sup>. Then for any <a href="/wiki/Real_number" title="Real number">real number</a> <span class="texhtml"><i>k</i> > 0</span>, </p><p><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 \Pr(|X-\mu |\geq k\sigma )\leq {\frac {1}{k^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mo>−</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mi>k</mi> <mi>σ</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(|X-\mu |\geq k\sigma )\leq {\frac {1}{k^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13787911b032508f2a54da8eb84750f331a70401" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:24.306ex; height:5.509ex;" alt="{\displaystyle \Pr(|X-\mu |\geq k\sigma )\leq {\frac {1}{k^{2}}}.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Proof_of_the_weak_law">Proof of the weak law</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Law_of_large_numbers&action=edit&section=10" title="Edit section: Proof of the weak law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given <i>X</i><sub>1</sub>, <i>X</i><sub>2</sub>, ... an infinite sequence of <a href="/wiki/I.i.d." class="mw-redirect" title="I.i.d.">i.i.d.</a> random variables 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="{\displaystyle E(X_{1})=E(X_{2})=\cdots =\mu <\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>⋯</mo> <mo>=</mo> <mi>μ</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(X_{1})=E(X_{2})=\cdots =\mu <\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88845308c97251d8af2d8f9f70d35428ed8c21eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.969ex; height:2.843ex;" alt="{\displaystyle E(X_{1})=E(X_{2})=\cdots =\mu <\infty }"></span>, we are interested in the convergence of the sample average </p><p><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 {\overline {X}}_{n}={\tfrac {1}{n}}(X_{1}+\cdots +X_{n}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo 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 stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}_{n}={\tfrac {1}{n}}(X_{1}+\cdots +X_{n}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a12cd0ecc4f2e2ad989599295c475d890ebd6f17" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.252ex; height:3.676ex;" alt="{\displaystyle {\overline {X}}_{n}={\tfrac {1}{n}}(X_{1}+\cdots +X_{n}).}"></span> </p><p>The weak law of large numbers states: </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 {\overline {X}}_{n}\ {\overset {P}{\rightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo stretchy="false">→</mo> <mi>P</mi> </mover> </mrow> <mtext> </mtext> <mi>μ</mi> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>when</mtext> </mrow> </mrow> <mtext> </mtext> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}_{n}\ {\overset {P}{\rightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e9cd779400133a1b72dc4ac5695c46b44f1740f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.737ex; height:4.176ex;" alt="{\displaystyle {\overline {X}}_{n}\ {\overset {P}{\rightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span>)</b></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Proof_using_Chebyshev's_inequality_assuming_finite_variance"><span id="Proof_using_Chebyshev.27s_inequality_assuming_finite_variance"></span>Proof using Chebyshev's inequality assuming finite variance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Law_of_large_numbers&action=edit&section=11" title="Edit section: Proof using Chebyshev's inequality assuming finite variance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This proof uses the assumption of finite <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 \operatorname {Var} (X_{i})=\sigma ^{2}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} (X_{i})=\sigma ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd1be36beafcc2454eaa20fb8e09a00cc2e2fa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.834ex; height:3.176ex;" alt="{\displaystyle \operatorname {Var} (X_{i})=\sigma ^{2}}"></span> (for all <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>). The independence of the random variables implies no correlation between them, and we have that </p><p><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 \operatorname {Var} ({\overline {X}}_{n})=\operatorname {Var} ({\tfrac {1}{n}}(X_{1}+\cdots +X_{n}))={\frac {1}{n^{2}}}\operatorname {Var} (X_{1}+\cdots +X_{n})={\frac {n\sigma ^{2}}{n^{2}}}={\frac {\sigma ^{2}}{n}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo 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 stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>Var</mi> <mo>⁡</mo> <mo 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 stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Var} ({\overline {X}}_{n})=\operatorname {Var} ({\tfrac {1}{n}}(X_{1}+\cdots +X_{n}))={\frac {1}{n^{2}}}\operatorname {Var} (X_{1}+\cdots +X_{n})={\frac {n\sigma ^{2}}{n^{2}}}={\frac {\sigma ^{2}}{n}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3afd10e29bec1e03a91c726c4ddbfd29ccdc40e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:77.461ex; height:6.009ex;" alt="{\displaystyle \operatorname {Var} ({\overline {X}}_{n})=\operatorname {Var} ({\tfrac {1}{n}}(X_{1}+\cdots +X_{n}))={\frac {1}{n^{2}}}\operatorname {Var} (X_{1}+\cdots +X_{n})={\frac {n\sigma ^{2}}{n^{2}}}={\frac {\sigma ^{2}}{n}}.}"></span> </p><p>The common mean μ of the sequence is the mean of the sample average: </p><p><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 E({\overline {X}}_{n})=\mu .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>μ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E({\overline {X}}_{n})=\mu .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca13475c69bebdfe0dc2aec57eb5168aa0602979" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.082ex; height:3.509ex;" alt="{\displaystyle E({\overline {X}}_{n})=\mu .}"></span> </p><p>Using <a href="/wiki/Chebyshev%27s_inequality" title="Chebyshev's inequality">Chebyshev's inequality</a> on <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 {\overline {X}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55a63d07fe7c766a92154413faf8456d31c29fca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.35ex; height:3.343ex;" alt="{\displaystyle {\overline {X}}_{n}}"></span> results in </p><p><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 \operatorname {P} (\left|{\overline {X}}_{n}-\mu \right|\geq \varepsilon )\leq {\frac {\sigma ^{2}}{n\varepsilon ^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow> <mo>|</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> </mrow> <mo>|</mo> </mrow> <mo>≥</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>n</mi> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {P} (\left|{\overline {X}}_{n}-\mu \right|\geq \varepsilon )\leq {\frac {\sigma ^{2}}{n\varepsilon ^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2000124280d7f9fb02e500ce5e990e0d72024245" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:24.573ex; height:6.009ex;" alt="{\displaystyle \operatorname {P} (\left|{\overline {X}}_{n}-\mu \right|\geq \varepsilon )\leq {\frac {\sigma ^{2}}{n\varepsilon ^{2}}}.}"></span> </p><p>This may be used to obtain the following: </p><p><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 \operatorname {P} (\left|{\overline {X}}_{n}-\mu \right|<\varepsilon )=1-\operatorname {P} (\left|{\overline {X}}_{n}-\mu \right|\geq \varepsilon )\geq 1-{\frac {\sigma ^{2}}{n\varepsilon ^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow> <mo>|</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> </mrow> <mo>|</mo> </mrow> <mo><</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow> <mo>|</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> </mrow> <mo>|</mo> </mrow> <mo>≥</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>n</mi> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {P} (\left|{\overline {X}}_{n}-\mu \right|<\varepsilon )=1-\operatorname {P} (\left|{\overline {X}}_{n}-\mu \right|\geq \varepsilon )\geq 1-{\frac {\sigma ^{2}}{n\varepsilon ^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e0fe094f804165281275d8f7b6d20ec7cf41838" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:52.137ex; height:6.009ex;" alt="{\displaystyle \operatorname {P} (\left|{\overline {X}}_{n}-\mu \right|<\varepsilon )=1-\operatorname {P} (\left|{\overline {X}}_{n}-\mu \right|\geq \varepsilon )\geq 1-{\frac {\sigma ^{2}}{n\varepsilon ^{2}}}.}"></span> </p><p>As <i>n</i> approaches infinity, the expression approaches 1. And by definition of <a href="/wiki/Convergence_in_probability" class="mw-redirect" title="Convergence in probability">convergence in probability</a>, we have obtained </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 {\overline {X}}_{n}\ {\overset {P}{\rightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo stretchy="false">→</mo> <mi>P</mi> </mover> </mrow> <mtext> </mtext> <mi>μ</mi> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>when</mtext> </mrow> </mrow> <mtext> </mtext> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}_{n}\ {\overset {P}{\rightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e9cd779400133a1b72dc4ac5695c46b44f1740f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.737ex; height:4.176ex;" alt="{\displaystyle {\overline {X}}_{n}\ {\overset {P}{\rightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span>)</b></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Proof_using_convergence_of_characteristic_functions">Proof using convergence of characteristic functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Law_of_large_numbers&action=edit&section=12" title="Edit section: Proof using convergence of characteristic functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By <a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a> for <a href="/wiki/Complex_function" class="mw-redirect" title="Complex function">complex functions</a>, the <a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a> of any random variable, <i>X</i>, with finite mean μ, can be written as </p><p><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 _{X}(t)=1+it\mu +o(t),\quad t\rightarrow 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mi>t</mi> <mi>μ</mi> <mo>+</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{X}(t)=1+it\mu +o(t),\quad t\rightarrow 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93f1b9f3d2a336effb48f074c5dbbc3138f2a25c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.183ex; height:2.843ex;" alt="{\displaystyle \varphi _{X}(t)=1+it\mu +o(t),\quad t\rightarrow 0.}"></span> </p><p>All <i>X</i><sub>1</sub>, <i>X</i><sub>2</sub>, ... have the same characteristic function, so we will simply denote this <i>φ</i><sub><i>X</i></sub>. </p><p>Among the basic properties of characteristic functions there are </p><p><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 _{{\frac {1}{n}}X}(t)=\varphi _{X}({\tfrac {t}{n}})\quad {\text{and}}\quad \varphi _{X+Y}(t)=\varphi _{X}(t)\varphi _{Y}(t)\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>t</mi> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>+</mo> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{{\frac {1}{n}}X}(t)=\varphi _{X}({\tfrac {t}{n}})\quad {\text{and}}\quad \varphi _{X+Y}(t)=\varphi _{X}(t)\varphi _{Y}(t)\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b3595e14a82149465f5dc152247e52d8bfa4291" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:50.925ex; height:4.176ex;" alt="{\displaystyle \varphi _{{\frac {1}{n}}X}(t)=\varphi _{X}({\tfrac {t}{n}})\quad {\text{and}}\quad \varphi _{X+Y}(t)=\varphi _{X}(t)\varphi _{Y}(t)\quad }"></span> if <i>X</i> and <i>Y</i> are independent. </p><p>These rules can be used to calculate 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 {\overline {X}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55a63d07fe7c766a92154413faf8456d31c29fca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.35ex; height:3.343ex;" alt="{\displaystyle {\overline {X}}_{n}}"></span> in terms of <i>φ</i><sub><i>X</i></sub>: </p><p><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 _{{\overline {X}}_{n}}(t)=\left[\varphi _{X}\left({t \over n}\right)\right]^{n}=\left[1+i\mu {t \over n}+o\left({t \over n}\right)\right]^{n}\,\rightarrow \,e^{it\mu },\quad {\text{as}}\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> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <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> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mi>o</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo stretchy="false">→</mo> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> <mi>μ</mi> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>as</mtext> </mrow> <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 _{{\overline {X}}_{n}}(t)=\left[\varphi _{X}\left({t \over n}\right)\right]^{n}=\left[1+i\mu {t \over n}+o\left({t \over n}\right)\right]^{n}\,\rightarrow \,e^{it\mu },\quad {\text{as}}\quad n\to \infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/244552d66f95379a21ce004941baeb45f7cccc1d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:71.659ex; height:6.176ex;" alt="{\displaystyle \varphi _{{\overline {X}}_{n}}(t)=\left[\varphi _{X}\left({t \over n}\right)\right]^{n}=\left[1+i\mu {t \over n}+o\left({t \over n}\right)\right]^{n}\,\rightarrow \,e^{it\mu },\quad {\text{as}}\quad n\to \infty .}"></span> </p><p>The limit <i>e</i><sup><i>itμ</i></sup> is the characteristic function of the constant random variable μ, and hence by the <a href="/wiki/L%C3%A9vy_continuity_theorem" class="mw-redirect" title="Lévy continuity theorem">Lévy continuity theorem</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 {\overline {X}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55a63d07fe7c766a92154413faf8456d31c29fca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.35ex; height:3.343ex;" alt="{\displaystyle {\overline {X}}_{n}}"></span> <a href="/wiki/Convergence_in_distribution" class="mw-redirect" title="Convergence in distribution">converges in distribution</a> to μ: </p><p><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 {\overline {X}}_{n}\,{\overset {\mathcal {D}}{\rightarrow }}\,\mu \qquad {\text{for}}\qquad n\to \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mover> </mrow> <mspace width="thinmathspace" /> <mi>μ</mi> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for</mtext> </mrow> <mspace width="2em" /> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}_{n}\,{\overset {\mathcal {D}}{\rightarrow }}\,\mu \qquad {\text{for}}\qquad n\to \infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d033bb000f88d5f5dcebd62f6f8804ccf17a63b4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.905ex; height:4.176ex;" alt="{\displaystyle {\overline {X}}_{n}\,{\overset {\mathcal {D}}{\rightarrow }}\,\mu \qquad {\text{for}}\qquad n\to \infty .}"></span> </p><p>μ is a constant, which implies that convergence in distribution to μ and convergence in probability to μ are equivalent (see <a href="/wiki/Convergence_of_random_variables" title="Convergence of random variables">Convergence of random variables</a>.) Therefore, </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 {\overline {X}}_{n}\ {\overset {P}{\rightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo stretchy="false">→</mo> <mi>P</mi> </mover> </mrow> <mtext> </mtext> <mi>μ</mi> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>when</mtext> </mrow> </mrow> <mtext> </mtext> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}_{n}\ {\overset {P}{\rightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e9cd779400133a1b72dc4ac5695c46b44f1740f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.737ex; height:4.176ex;" alt="{\displaystyle {\overline {X}}_{n}\ {\overset {P}{\rightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span>)</b></td></tr></tbody></table> <p>This shows that the sample mean converges in probability to the derivative of the characteristic function at the origin, as long as the latter exists. </p> <div class="mw-heading mw-heading2"><h2 id="Proof_of_the_strong_law">Proof of the strong law</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Law_of_large_numbers&action=edit&section=13" title="Edit section: Proof of the strong law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We give a relatively simple proof of the strong law under the assumptions that 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 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> </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/af4a0955af42beb5f85aa05fb8c07abedc13990d" 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="{\displaystyle X_{i}}"></span> are <a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">iid</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 {\mathbb {E} }[X_{i}]=:\mu <\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <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> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {E} }[X_{i}]=:\mu <\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/945172be4f9585c1118ce2188617324caa19fdcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.137ex; height:2.843ex;" alt="{\displaystyle {\mathbb {E} }[X_{i}]=:\mu <\infty }"></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 \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> </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/e330828bee090146fc43261d66285f9a516b9ce0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.256ex; height:3.176ex;" alt="{\displaystyle \operatorname {Var} (X_{i})=\sigma ^{2}<\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 {\mathbb {E} }[X_{i}^{4}]=:\tau <\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo stretchy="false">]</mo> <mo>=:</mo> <mi>τ</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {E} }[X_{i}^{4}]=:\tau <\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/089cf58c0dc1335eb8078f5ea60b0f37e6ba7080" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.265ex; height:3.176ex;" alt="{\displaystyle {\mathbb {E} }[X_{i}^{4}]=:\tau <\infty }"></span>. </p><p>Let us first note that without loss of generality we can assume 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="{\displaystyle \mu =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3753282c0ad2ea1e7d63f39425efd13c37da3169" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.676ex;" alt="{\displaystyle \mu =0}"></span> by centering. In this case, the strong law says that </p><p><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 \Pr \!\left(\lim _{n\to \infty }{\overline {X}}_{n}=0\right)=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow> <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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr \!\left(\lim _{n\to \infty }{\overline {X}}_{n}=0\right)=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c599f8a72d0f457f8d823e4f8f54867eb12830e8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.448ex; height:4.843ex;" alt="{\displaystyle \Pr \!\left(\lim _{n\to \infty }{\overline {X}}_{n}=0\right)=1,}"></span> or <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 \Pr \left(\omega :\lim _{n\to \infty }{\frac {S_{n}(\omega )}{n}}=0\right)=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mrow> <mo>(</mo> <mrow> <mi>ω</mi> <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> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr \left(\omega :\lim _{n\to \infty }{\frac {S_{n}(\omega )}{n}}=0\right)=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d57823168b1ecb026e9cf31027c40ce6ca6833e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.249ex; height:6.343ex;" alt="{\displaystyle \Pr \left(\omega :\lim _{n\to \infty }{\frac {S_{n}(\omega )}{n}}=0\right)=1.}"></span> It is equivalent to show 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 \Pr \left(\omega :\lim _{n\to \infty }{\frac {S_{n}(\omega )}{n}}\neq 0\right)=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mrow> <mo>(</mo> <mrow> <mi>ω</mi> <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> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>≠</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr \left(\omega :\lim _{n\to \infty }{\frac {S_{n}(\omega )}{n}}\neq 0\right)=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14cf42171fdbad285109ec0f1997b143c7abc9b0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.249ex; height:6.343ex;" alt="{\displaystyle \Pr \left(\omega :\lim _{n\to \infty }{\frac {S_{n}(\omega )}{n}}\neq 0\right)=0,}"></span> Note 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 }{\frac {S_{n}(\omega )}{n}}\neq 0\iff \exists \epsilon >0,\left|{\frac {S_{n}(\omega )}{n}}\right|\geq \epsilon \ {\mbox{infinitely often}},}"> <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> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>≠</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">∃</mi> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>|</mo> </mrow> <mo>≥</mo> <mi>ϵ</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>infinitely often</mtext> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {S_{n}(\omega )}{n}}\neq 0\iff \exists \epsilon >0,\left|{\frac {S_{n}(\omega )}{n}}\right|\geq \epsilon \ {\mbox{infinitely often}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de5dbfe24422c35a5df6c686e742c12863747944" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:58.116ex; height:6.509ex;" alt="{\displaystyle \lim _{n\to \infty }{\frac {S_{n}(\omega )}{n}}\neq 0\iff \exists \epsilon >0,\left|{\frac {S_{n}(\omega )}{n}}\right|\geq \epsilon \ {\mbox{infinitely often}},}"></span> and thus to prove the strong law we need to show 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="{\displaystyle \epsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}"></span>, we have <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 \Pr \left(\omega :|S_{n}(\omega )|\geq n\epsilon {\mbox{ infinitely often}}\right)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mrow> <mo>(</mo> <mrow> <mi>ω</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mi>n</mi> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> infinitely often</mtext> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr \left(\omega :|S_{n}(\omega )|\geq n\epsilon {\mbox{ infinitely often}}\right)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7af3219b7ea6de794a8bc7784435486b2ed858c0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.925ex; height:2.843ex;" alt="{\displaystyle \Pr \left(\omega :|S_{n}(\omega )|\geq n\epsilon {\mbox{ infinitely often}}\right)=0.}"></span> Define the events <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 A_{n}=\{\omega :|S_{n}|\geq n\epsilon \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>ω</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mi>n</mi> <mi>ϵ</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}=\{\omega :|S_{n}|\geq n\epsilon \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39e4829b6609562e31eca4ee6428781f77abd59b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.142ex; height:2.843ex;" alt="{\displaystyle A_{n}=\{\omega :|S_{n}|\geq n\epsilon \}}"></span>, and if we can show 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 \sum _{n=1}^{\infty }\Pr(A_{n})<\infty ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }\Pr(A_{n})<\infty ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46b48c566f546abc7c5aacf669fdad4f5bbc2a61" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.076ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }\Pr(A_{n})<\infty ,}"></span> then the Borel-Cantelli Lemma implies the result. So let us estimate <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 \Pr(A_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(A_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a5de6a709912fdd89354029efdec49e29c675f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.265ex; height:2.843ex;" alt="{\displaystyle \Pr(A_{n})}"></span>. </p><p>We compute <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 {\mathbb {E} }[S_{n}^{4}]={\mathbb {E} }\left[\left(\sum _{i=1}^{n}X_{i}\right)^{4}\right]={\mathbb {E} }\left[\sum _{1\leq i,j,k,l\leq n}X_{i}X_{j}X_{k}X_{l}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mrow> <mo>[</mo> <msup> <mrow> <mo>(</mo> <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> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mrow> <mo>[</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>l</mi> <mo>≤</mo> <mi>n</mi> </mrow> </munder> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {E} }[S_{n}^{4}]={\mathbb {E} }\left[\left(\sum _{i=1}^{n}X_{i}\right)^{4}\right]={\mathbb {E} }\left[\sum _{1\leq i,j,k,l\leq n}X_{i}X_{j}X_{k}X_{l}\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fed903db3f4b0526dcf5ee44a967bac45eeb9ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:53.54ex; height:8.509ex;" alt="{\displaystyle {\mathbb {E} }[S_{n}^{4}]={\mathbb {E} }\left[\left(\sum _{i=1}^{n}X_{i}\right)^{4}\right]={\mathbb {E} }\left[\sum _{1\leq i,j,k,l\leq n}X_{i}X_{j}X_{k}X_{l}\right].}"></span> We first claim that every term of the form <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}^{3}X_{j},X_{i}^{2}X_{j}X_{k},X_{i}X_{j}X_{k}X_{l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}^{3}X_{j},X_{i}^{2}X_{j}X_{k},X_{i}X_{j}X_{k}X_{l}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4088db316a59e3852c28284fe26d6acb1493d9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.069ex; height:3.176ex;" alt="{\displaystyle X_{i}^{3}X_{j},X_{i}^{2}X_{j}X_{k},X_{i}X_{j}X_{k}X_{l}}"></span> where all subscripts are distinct, must have zero expectation. This is because <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 {E} }[X_{i}^{3}X_{j}]={\mathbb {E} }[X_{i}^{3}]{\mathbb {E} }[X_{j}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {E} }[X_{i}^{3}X_{j}]={\mathbb {E} }[X_{i}^{3}]{\mathbb {E} }[X_{j}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45c42392e59a0113c14527a2256b4d2c6864c7ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.401ex; height:3.176ex;" alt="{\displaystyle {\mathbb {E} }[X_{i}^{3}X_{j}]={\mathbb {E} }[X_{i}^{3}]{\mathbb {E} }[X_{j}]}"></span> by independence, and the last term is zero --- and similarly for the other terms. Therefore the only terms in the sum with nonzero expectation are <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 {E} }[X_{i}^{4}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {E} }[X_{i}^{4}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d33c75b5a66fea493222d077bb54d98f960bf082" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.895ex; height:3.176ex;" alt="{\displaystyle {\mathbb {E} }[X_{i}^{4}]}"></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 {\mathbb {E} }[X_{i}^{2}X_{j}^{2}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {E} }[X_{i}^{2}X_{j}^{2}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d22184db5e10e679e4d366aac99b4ca70205d99f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.946ex; height:3.509ex;" alt="{\displaystyle {\mathbb {E} }[X_{i}^{2}X_{j}^{2}]}"></span>. Since 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 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> </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/af4a0955af42beb5f85aa05fb8c07abedc13990d" 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="{\displaystyle X_{i}}"></span> are identically distributed, all of these are the same, and moreover <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 {E} }[X_{i}^{2}X_{j}^{2}]=({\mathbb {E} }[X_{i}^{2}])^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">]</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {E} }[X_{i}^{2}X_{j}^{2}]=({\mathbb {E} }[X_{i}^{2}])^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2296e8ef46d13f64a00609c21c838e842d739a69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.803ex; height:3.676ex;" alt="{\displaystyle {\mathbb {E} }[X_{i}^{2}X_{j}^{2}]=({\mathbb {E} }[X_{i}^{2}])^{2}}"></span>. </p><p>There are <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> terms of the form <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 {E} }[X_{i}^{4}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {E} }[X_{i}^{4}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d33c75b5a66fea493222d077bb54d98f960bf082" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.895ex; height:3.176ex;" alt="{\displaystyle {\mathbb {E} }[X_{i}^{4}]}"></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 3n(n-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3n(n-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14201e5d35ae09a3b50026d24996dd73e90e7271" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.764ex; height:2.843ex;" alt="{\displaystyle 3n(n-1)}"></span> terms of the form <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 {E} }[X_{i}^{2}])^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">]</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathbb {E} }[X_{i}^{2}])^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95021f2378d75a081aae6398232a2ac38ae0f7a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.759ex; height:3.343ex;" alt="{\displaystyle ({\mathbb {E} }[X_{i}^{2}])^{2}}"></span>, and so <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 {\mathbb {E} }[S_{n}^{4}]=n\tau +3n(n-1)\sigma ^{4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo stretchy="false">]</mo> <mo>=</mo> <mi>n</mi> <mi>τ</mi> <mo>+</mo> <mn>3</mn> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {E} }[S_{n}^{4}]=n\tau +3n(n-1)\sigma ^{4}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0800261defee5ae3d5e94f20fc83fe231a277fee" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.819ex; height:3.176ex;" alt="{\displaystyle {\mathbb {E} }[S_{n}^{4}]=n\tau +3n(n-1)\sigma ^{4}.}"></span> Note that the right-hand side is a quadratic polynomial 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 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>, and as such there exists 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 C>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c84d4126c6df243734f9355927c026df6b0d3859" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.027ex; height:2.176ex;" alt="{\displaystyle C>0}"></span> such 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="{\displaystyle {\mathbb {E} }[S_{n}^{4}]\leq Cn^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo stretchy="false">]</mo> <mo>≤</mo> <mi>C</mi> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {E} }[S_{n}^{4}]\leq Cn^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93c1f1e646da5df999c4c952025d757291bd2f0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.801ex; height:3.176ex;" alt="{\displaystyle {\mathbb {E} }[S_{n}^{4}]\leq Cn^{2}}"></span> 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 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> sufficiently large. By Markov, <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 \Pr(|S_{n}|\geq n\epsilon )\leq {\frac {1}{(n\epsilon )^{4}}}{\mathbb {E} }[S_{n}^{4}]\leq {\frac {C}{\epsilon ^{4}n^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mi>n</mi> <mi>ϵ</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mi>ϵ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo stretchy="false">]</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mrow> <msup> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(|S_{n}|\geq n\epsilon )\leq {\frac {1}{(n\epsilon )^{4}}}{\mathbb {E} }[S_{n}^{4}]\leq {\frac {C}{\epsilon ^{4}n^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb57fec48553c08499aded735ca22333704137f2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.331ex; height:6.176ex;" alt="{\displaystyle \Pr(|S_{n}|\geq n\epsilon )\leq {\frac {1}{(n\epsilon )^{4}}}{\mathbb {E} }[S_{n}^{4}]\leq {\frac {C}{\epsilon ^{4}n^{2}}},}"></span> 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 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> sufficiently large, and therefore this series is summable. Since this holds for any <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 \epsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}"></span>, we have established the Strong LLN. </p><p><br /> Another proof was given by Etemadi.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p><p>For a proof without the added assumption of a finite fourth moment, see Section 22 of Billingsley.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Consequences">Consequences</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Law_of_large_numbers&action=edit&section=14" title="Edit section: Consequences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The law of large numbers provides an expectation of an unknown distribution from a realization of the sequence, but also any feature of the <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a>.<sup id="cite_ref-:0_1-3" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> By applying <a href="/wiki/Borel%27s_law_of_large_numbers" class="mw-redirect" title="Borel's law of large numbers">Borel's law of large numbers</a>, one could easily obtain the probability mass function. For each event in the objective probability mass function, one could approximate the probability of the event's occurrence with the proportion of times that any specified event occurs. The larger the number of repetitions, the better the approximation. As for the continuous 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 C=(a-h,a+h]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>h</mi> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=(a-h,a+h]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb64a0e36558e4189ac0f21fef2370c4649aa542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.269ex; height:2.843ex;" alt="{\displaystyle C=(a-h,a+h]}"></span>, for small positive h. Thus, for large n: </p><p><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 {N_{n}(C)}{n}}\thickapprox p=P(X\in C)=\int _{a-h}^{a+h}f(x)\,dx\thickapprox 2hf(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mrow> <mo class="MJX-variant">≈</mo> <mi>p</mi> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>∈</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>−</mo> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>+</mo> <mi>h</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo class="MJX-variant">≈</mo> <mn>2</mn> <mi>h</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {N_{n}(C)}{n}}\thickapprox p=P(X\in C)=\int _{a-h}^{a+h}f(x)\,dx\thickapprox 2hf(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/327d2962eae278b2988f5a0051cd6c31342d756a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.549ex; height:6.509ex;" alt="{\displaystyle {\frac {N_{n}(C)}{n}}\thickapprox p=P(X\in C)=\int _{a-h}^{a+h}f(x)\,dx\thickapprox 2hf(a)}"></span> </p><p>With this method, one can cover the whole x-axis with a grid (with grid size 2h) and obtain a bar graph which is called a <a href="/wiki/Histogram" title="Histogram">histogram</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Law_of_large_numbers&action=edit&section=15" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One application of the LLN is an important method of approximation known as the <a href="/wiki/Monte_Carlo_method" title="Monte Carlo method">Monte Carlo method</a>,<sup id="cite_ref-:1_3-5" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> which uses a random sampling of numbers to approximate numerical results. The algorithm to compute an integral of f(x) on an interval [a,b] is as follows:<sup id="cite_ref-:1_3-6" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <ol><li>Simulate uniform random variables X<sub>1</sub>, X<sub>2</sub>, ..., X<sub>n</sub> which can be done using a software, and use a random number table that gives U<sub>1</sub>, U<sub>2</sub>, ..., U<sub>n</sub> independent and identically distributed (i.i.d.) random variables on [0,1]. Then let X<sub>i</sub> = a+(b - a)U<sub>i</sub> for i= 1, 2, ..., n. Then X<sub>1</sub>, X<sub>2</sub>, ..., X<sub>n</sub> are independent and identically distributed uniform random variables on [a, b].</li> <li>Evaluate f(X<sub>1</sub>), f(X<sub>2</sub>), ..., f(X<sub>n</sub>)</li> <li>Take the average of f(X<sub>1</sub>), f(X<sub>2</sub>), ..., f(X<sub>n</sub>) by computing <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 (b-a){\tfrac {f(X_{1})+f(X_{2})+...+f(X_{n})}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (b-a){\tfrac {f(X_{1})+f(X_{2})+...+f(X_{n})}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a45523b08f4fe532bc8cdaaf0dc0fbde8d1b7f5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.181ex; height:4.009ex;" alt="{\displaystyle (b-a){\tfrac {f(X_{1})+f(X_{2})+...+f(X_{n})}{n}}}"></span> and then by the Strong Law of Large Numbers, this converges 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 (b-a)E(f(X_{1}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (b-a)E(f(X_{1}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86eb0bdc5f2274eb7cd0e80103e43d960cd11f78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.528ex; height:2.843ex;" alt="{\displaystyle (b-a)E(f(X_{1}))}"></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 (b-a)\int _{a}^{b}f(x){\tfrac {1}{b-a}}{dx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (b-a)\int _{a}^{b}f(x){\tfrac {1}{b-a}}{dx}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cae3ae0671585ecb8acee5e2aab1b98c32c8b8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.706ex; height:6.343ex;" alt="{\displaystyle (b-a)\int _{a}^{b}f(x){\tfrac {1}{b-a}}{dx}}"></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 \int _{a}^{b}f(x){dx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x){dx}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4733c32f1b636e624952773ff535f26d99633977" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.752ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x){dx}}"></span></li></ol> <p>We can find the integral 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 f(x)=cos^{2}(x){\sqrt {x^{3}+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> <mi>o</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=cos^{2}(x){\sqrt {x^{3}+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67896af5c8bed23c23e094e8a9ddd43ffdcef445" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.645ex; height:3.509ex;" alt="{\displaystyle f(x)=cos^{2}(x){\sqrt {x^{3}+1}}}"></span> on [-1,2]. Using traditional methods to compute this integral is very difficult, so the Monte Carlo method can be used here.<sup id="cite_ref-:1_3-7" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Using the above algorithm, we get </p><p><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 \int _{-1}^{2}f(x){dx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-1}^{2}f(x){dx}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab97dd478cf01af3eb324a4f9ffca9b16e43edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.976ex; height:6.343ex;" alt="{\displaystyle \int _{-1}^{2}f(x){dx}}"></span> = 0.905 when n=25 </p><p>and </p><p><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 \int _{-1}^{2}f(x){dx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-1}^{2}f(x){dx}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab97dd478cf01af3eb324a4f9ffca9b16e43edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.976ex; height:6.343ex;" alt="{\displaystyle \int _{-1}^{2}f(x){dx}}"></span> = 1.028 when n=250 </p><p>We observe that as n increases, the numerical value also increases. When we get the actual results for the integral we get </p><p><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 \int _{-1}^{2}f(x){dx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-1}^{2}f(x){dx}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab97dd478cf01af3eb324a4f9ffca9b16e43edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.976ex; height:6.343ex;" alt="{\displaystyle \int _{-1}^{2}f(x){dx}}"></span> = 1.000194 </p><p>When the LLN was used, the approximation of the integral was closer to its true value, and thus more accurate.<sup id="cite_ref-:1_3-8" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Another example is the integration of <big>f(x) =</big> <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 {e^{x}-1}{e-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>e</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {e^{x}-1}{e-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab562594d5c63e0349b9b47abd6357738057766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.095ex; height:5.343ex;" alt="{\displaystyle {\frac {e^{x}-1}{e-1}}}"></span> on [0,1].<sup id="cite_ref-:2_34-0" class="reference"><a href="#cite_note-:2-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> Using the Monte Carlo method and the LLN, we can see that as the number of samples increases, the numerical value gets closer to 0.4180233.<sup id="cite_ref-:2_34-1" class="reference"><a href="#cite_note-:2-34"><span class="cite-bracket">[</span>34<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=Law_of_large_numbers&action=edit&section=16" 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/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li> <li><a href="/wiki/Infinite_monkey_theorem" title="Infinite monkey theorem">Infinite monkey theorem</a></li> <li><a href="/wiki/A_Treatise_on_Probability" title="A Treatise on Probability">Keynes' Treatise on Probability</a></li> <li><a href="/wiki/Law_of_averages" title="Law of averages">Law of averages</a></li> <li><a href="/wiki/Law_of_the_iterated_logarithm" title="Law of the iterated logarithm">Law of the iterated logarithm</a></li> <li><a href="/wiki/Law_of_truly_large_numbers" title="Law of truly large numbers">Law of truly large numbers</a></li> <li><a href="/wiki/Lindy_effect" title="Lindy effect">Lindy effect</a></li> <li><a href="/wiki/Regression_toward_the_mean" title="Regression toward the mean">Regression toward the mean</a></li> <li><a href="/wiki/Sortition" title="Sortition">Sortition</a></li> <li><a href="/wiki/Strong_law_of_small_numbers" title="Strong law of small numbers">Strong law of small numbers</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=Law_of_large_numbers&action=edit&section=17" 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 reflist-columns-2"> <ol class="references"> <li id="cite_note-:0-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:0_1-3"><sup><i><b>d</b></i></sup></a></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="CITEREFDekking2005" class="citation book cs1">Dekking, Michel (2005). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/modernintroducti00fmde"><i>A Modern Introduction to Probability and Statistics</i></a></span>. Springer. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/modernintroducti00fmde/page/n191">181</a>–190. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781852338961" title="Special:BookSources/9781852338961"><bdi>9781852338961</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+Modern+Introduction+to+Probability+and+Statistics&rft.pages=181-190&rft.pub=Springer&rft.date=2005&rft.isbn=9781852338961&rft.aulast=Dekking&rft.aufirst=Michel&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmodernintroducti00fmde&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaw+of+large+numbers" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYaoGao2016" class="citation journal cs1">Yao, Kai; Gao, Jinwu (2016). 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(2005). <i>A modern introduction to probability and statistics: understanding why and how</i>. Springer texts in statistics. 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Springer. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/modernintroducti00fmde/page/n102">92</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781852338961" title="Special:BookSources/9781852338961"><bdi>9781852338961</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+Modern+Introduction+to+Probability+and+Statistics&rft.pages=92&rft.pub=Springer&rft.date=2005&rft.isbn=9781852338961&rft.aulast=Dekking&rft.aufirst=Michel&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmodernintroducti00fmde&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaw+of+large+numbers" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDekking2005" class="citation book cs1">Dekking, Michel (2005). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/modernintroducti00fmde"><i>A Modern Introduction to Probability and Statistics</i></a></span>. Springer. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/modernintroducti00fmde/page/n74">63</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781852338961" title="Special:BookSources/9781852338961"><bdi>9781852338961</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+Modern+Introduction+to+Probability+and+Statistics&rft.pages=63&rft.pub=Springer&rft.date=2005&rft.isbn=9781852338961&rft.aulast=Dekking&rft.aufirst=Michel&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmodernintroducti00fmde&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaw+of+large+numbers" 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="CITEREFPitmanWilliams1967" class="citation journal cs1">Pitman, E. J. G.; Williams, E. J. (1967). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177698885">"Cauchy-Distributed Functions of Cauchy Variates"</a>. <i>The Annals of Mathematical Statistics</i>. <b>38</b> (3): 916–918. <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%2Faoms%2F1177698885">10.1214/aoms/1177698885</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-4851">0003-4851</a>. <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/2239008">2239008</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Annals+of+Mathematical+Statistics&rft.atitle=Cauchy-Distributed+Functions+of+Cauchy+Variates&rft.volume=38&rft.issue=3&rft.pages=916-918&rft.date=1967&rft.issn=0003-4851&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2239008%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1214%2Faoms%2F1177698885&rft.aulast=Pitman&rft.aufirst=E.+J.+G.&rft.au=Williams%2C+E.+J.&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faoms%252F1177698885&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaw+of+large+numbers" 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="CITEREFMlodinow2008" class="citation book cs1">Mlodinow, L. (2008). <i>The Drunkard's Walk</i>. New York: Random House. p. 50.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Drunkard%27s+Walk&rft.place=New+York&rft.pages=50&rft.pub=Random+House&rft.date=2008&rft.aulast=Mlodinow&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaw+of+large+numbers" 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="CITEREFBernoulli1713" class="citation book cs1 cs1-prop-foreign-lang-source">Bernoulli, Jakob (1713). "4". <i>Ars Conjectandi: Usum & Applicationem Praecedentis Doctrinae in Civilibus, Moralibus & Oeconomicis</i> (in Latin). Translated by Sheynin, Oscar.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=4&rft.btitle=Ars+Conjectandi%3A+Usum+%26+Applicationem+Praecedentis+Doctrinae+in+Civilibus%2C+Moralibus+%26+Oeconomicis&rft.date=1713&rft.aulast=Bernoulli&rft.aufirst=Jakob&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaw+of+large+numbers" 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">Poisson names the "law of large numbers" (<span title="French-language text"><i lang="fr">la loi des grands nombres</i></span>) in: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoisson1837" class="citation book cs1 cs1-prop-foreign-lang-source">Poisson, S. D. (1837). <i>Probabilité des jugements en matière criminelle et en matière civile, précédées des règles générales du calcul des probabilitiés</i> (in French). Paris, France: Bachelier. p. <a rel="nofollow" class="external text" href="https://archive.org/details/recherchessurla02poisgoog/page/n30">7</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probabilit%C3%A9+des+jugements+en+mati%C3%A8re+criminelle+et+en+mati%C3%A8re+civile%2C+pr%C3%A9c%C3%A9d%C3%A9es+des+r%C3%A8gles+g%C3%A9n%C3%A9rales+du+calcul+des+probabiliti%C3%A9s&rft.place=Paris%2C+France&rft.pages=7&rft.pub=Bachelier&rft.date=1837&rft.aulast=Poisson&rft.aufirst=S.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaw+of+large+numbers" class="Z3988"></span> He attempts a two-part proof of the law on pp. 139–143 and pp. 277 ff.</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="CITEREFHacking1983" class="citation journal cs1">Hacking, Ian (1983). 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Springer: 119–122. <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/10.1007%2FBF01013465">10.1007/BF01013465</a></span>. <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:122166046">122166046</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Zeitschrift+f%C3%BCr+Wahrscheinlichkeitstheorie+und+verwandte+Gebiete&rft.atitle=An+elementary+proof+of+the+strong+law+of+large+numbers&rft.volume=55&rft.pages=119-122&rft.date=1981&rft_id=info%3Adoi%2F10.1007%2FBF01013465&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122166046%23id-name%3DS2CID&rft.aulast=Etemadi&rft.aufirst=Nasrollah&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252FBF01013465&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaw+of+large+numbers" class="Z3988"></span></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBillingsley1979" class="citation book cs1">Billingsley, Patrick (1979). <i>Probability and Measure</i>.</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.date=1979&rft.aulast=Billingsley&rft.aufirst=Patrick&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaw+of+large+numbers" class="Z3988"></span></span> </li> <li id="cite_note-:2-34"><span class="mw-cite-backlink">^ <a href="#cite_ref-:2_34-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:2_34-1"><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="CITEREFReiter2008" class="citation cs2">Reiter, Detlev (2008), Fehske, H.; Schneider, R.; Weiße, A. (eds.), <a rel="nofollow" class="external text" href="http://link.springer.com/10.1007/978-3-540-74686-7_3">"The Monte Carlo Method, an Introduction"</a>, <i>Computational Many-Particle Physics</i>, Lecture Notes in Physics, vol. 739, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 63–78, <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%2F978-3-540-74686-7_3">10.1007/978-3-540-74686-7_3</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-74685-0" title="Special:BookSources/978-3-540-74685-0"><bdi>978-3-540-74685-0</bdi></a><span class="reference-accessdate">, retrieved <span class="nowrap">2023-12-08</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computational+Many-Particle+Physics&rft.atitle=The+Monte+Carlo+Method%2C+an+Introduction&rft.volume=739&rft.pages=63-78&rft.date=2008&rft_id=info%3Adoi%2F10.1007%2F978-3-540-74686-7_3&rft.isbn=978-3-540-74685-0&rft.aulast=Reiter&rft.aufirst=Detlev&rft_id=http%3A%2F%2Flink.springer.com%2F10.1007%2F978-3-540-74686-7_3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaw+of+large+numbers" 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=Law_of_large_numbers&action=edit&section=18" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrimmettStirzaker1992" class="citation book cs1">Grimmett, G. R.; Stirzaker, D. R. (1992). <i>Probability and Random Processes</i> (2nd ed.). Oxford: Clarendon Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-853665-8" title="Special:BookSources/0-19-853665-8"><bdi>0-19-853665-8</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+Random+Processes&rft.place=Oxford&rft.edition=2nd&rft.pub=Clarendon+Press&rft.date=1992&rft.isbn=0-19-853665-8&rft.aulast=Grimmett&rft.aufirst=G.+R.&rft.au=Stirzaker%2C+D.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaw+of+large+numbers" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDurrett1995" class="citation book cs1">Durrett, Richard (1995). <i>Probability: Theory and Examples</i> (2nd ed.). Duxbury Press.</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=2nd&rft.pub=Duxbury+Press&rft.date=1995&rft.aulast=Durrett&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaw+of+large+numbers" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartin_Jacobsen1992" class="citation book cs1 cs1-prop-foreign-lang-source">Martin Jacobsen (1992). <i>Videregående Sandsynlighedsregning</i> [<i>Advanced Probability Theory</i>] (in Danish) (3rd ed.). Copenhagen: HCØ-tryk. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/87-91180-71-6" title="Special:BookSources/87-91180-71-6"><bdi>87-91180-71-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Videreg%C3%A5ende+Sandsynlighedsregning&rft.place=Copenhagen&rft.edition=3rd&rft.pub=HC%C3%98-tryk&rft.date=1992&rft.isbn=87-91180-71-6&rft.au=Martin+Jacobsen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaw+of+large+numbers" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLoève1977" class="citation book cs1">Loève, Michel (1977). <i>Probability theory 1</i> (4th ed.). Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability+theory+1&rft.edition=4th&rft.pub=Springer&rft.date=1977&rft.aulast=Lo%C3%A8ve&rft.aufirst=Michel&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaw+of+large+numbers" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeweyMcFadden1994" class="citation book cs1">Newey, Whitney K.; <a href="/wiki/Daniel_McFadden" title="Daniel McFadden">McFadden, Daniel</a> (1994). "36". <i>Large sample estimation and hypothesis testing</i>. Handbook of econometrics. Vol. IV. Elsevier Science. pp. 2111–2245.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=36&rft.btitle=Large+sample+estimation+and+hypothesis+testing&rft.series=Handbook+of+econometrics&rft.pages=2111-2245&rft.pub=Elsevier+Science&rft.date=1994&rft.aulast=Newey&rft.aufirst=Whitney+K.&rft.au=McFadden%2C+Daniel&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaw+of+large+numbers" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoss2009" class="citation book cs1">Ross, Sheldon (2009). <i>A first course in probability</i> (8th ed.). Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-603313-4" title="Special:BookSources/978-0-13-603313-4"><bdi>978-0-13-603313-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+first+course+in+probability&rft.edition=8th&rft.pub=Prentice+Hall&rft.date=2009&rft.isbn=978-0-13-603313-4&rft.aulast=Ross&rft.aufirst=Sheldon&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaw+of+large+numbers" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSenSinger1993" class="citation book cs1">Sen, P. K; Singer, J. M. (1993). <i>Large sample methods in statistics</i>. Chapman & Hall.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Large+sample+methods+in+statistics&rft.pub=Chapman+%26+Hall&rft.date=1993&rft.aulast=Sen&rft.aufirst=P.+K&rft.au=Singer%2C+J.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaw+of+large+numbers" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSeneta2013" class="citation journal cs1"><a href="/wiki/Eugene_Seneta" title="Eugene Seneta">Seneta, Eugene</a> (2013). "A Tricentenary history of the Law of Large Numbers". <i>Bernoulli</i>. <b>19</b> (4): 1088–1121. <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/1309.6488">1309.6488</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/10.3150%2F12-BEJSP12">10.3150/12-BEJSP12</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:88520834">88520834</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bernoulli&rft.atitle=A+Tricentenary+history+of+the+Law+of+Large+Numbers&rft.volume=19&rft.issue=4&rft.pages=1088-1121&rft.date=2013&rft_id=info%3Aarxiv%2F1309.6488&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A88520834%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.3150%2F12-BEJSP12&rft.aulast=Seneta&rft.aufirst=Eugene&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaw+of+large+numbers" class="Z3988"></span></li></ul> </div> <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=Law_of_large_numbers&action=edit&section=19" title="Edit section: External links"><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 class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Law_of_large_numbers">"Law of large numbers"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Law+of+large+numbers&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DLaw_of_large_numbers&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaw+of+large+numbers" class="Z3988"></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Weak_Law_of_Large_Numbers"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/WeakLawofLargeNumbers.html">"Weak Law of Large Numbers"</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=Weak+Law+of+Large+Numbers&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FWeakLawofLargeNumbers.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaw+of+large+numbers" class="Z3988"></span></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Strong_Law_of_Large_Numbers"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/StrongLawofLargeNumbers.html">"Strong Law of Large Numbers"</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=Strong+Law+of+Large+Numbers&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FStrongLawofLargeNumbers.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALaw+of+large+numbers" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20081110071309/http://animation.yihui.name/prob:law_of_large_numbers">Animations for the Law of Large Numbers</a> by Yihui Xie using the <a href="/wiki/R_(programming_language)" title="R (programming language)">R</a> package <a rel="nofollow" class="external text" href="https://cran.r-project.org/package=animation">animation</a></li> <li><a rel="nofollow" class="external text" href="http://www.businessinsider.com/law-of-large-numbers-tim-cook-2015-2">Apple CEO Tim Cook said something that would make statisticians cringe</a>. "We don't believe in such laws as laws of large numbers. This is sort of, uh, old dogma, I think, that was cooked up by somebody [..]" said Tim Cook and while: "However, the law of large numbers has nothing to do with large companies, large revenues, or large growth rates. The law of large numbers is a fundamental concept in probability theory and statistics, tying together theoretical probabilities that we can calculate to the actual outcomes of experiments that we empirically perform.<i> explained <a href="/wiki/Business_Insider" title="Business Insider">Business Insider</a></i></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 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Law of large numbers - Wikipedia