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<span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Skorokhod_space" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Skorokhod_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Skorokhod space</span> </div> </a> <ul id="toc-Skorokhod_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties_of_Skorokhod_space" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties_of_Skorokhod_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Properties of Skorokhod space</span> </div> </a> <button aria-controls="toc-Properties_of_Skorokhod_space-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 Properties of Skorokhod space subsection</span> </button> <ul id="toc-Properties_of_Skorokhod_space-sublist" class="vector-toc-list"> <li id="toc-Generalization_of_the_uniform_topology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalization_of_the_uniform_topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Generalization of the uniform topology</span> </div> </a> <ul id="toc-Generalization_of_the_uniform_topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Completeness" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Completeness"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Completeness</span> </div> </a> <ul id="toc-Completeness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Separability" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Separability"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Separability</span> </div> </a> <ul id="toc-Separability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tightness_in_Skorokhod_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tightness_in_Skorokhod_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Tightness in Skorokhod space</span> </div> </a> <ul id="toc-Tightness_in_Skorokhod_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_and_topological_structure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_and_topological_structure"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Algebraic and topological structure</span> </div> </a> <ul id="toc-Algebraic_and_topological_structure-sublist" class="vector-toc-list"> </ul> </li> </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">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-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">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main 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class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Càdlàg</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. 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href="https://es.wikipedia.org/wiki/C%C3%A0dl%C3%A0g" title="Càdlàg – Spanish" lang="es" hreflang="es" data-title="Càdlàg" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D8%A8%D8%B9_C%C3%A0dl%C3%A0g" title="تابع Càdlàg – Persian" lang="fa" hreflang="fa" data-title="تابع Càdlàg" 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/C%C3%A0dl%C3%A0g" title="Càdlàg – French" lang="fr" hreflang="fr" data-title="Càdlàg" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8A%A4%EC%BD%94%EB%A1%9C%ED%98%B8%EB%93%9C_%EA%B3%B5%EA%B0%84" title="스코로호드 공간 – Korean" lang="ko" hreflang="ko" data-title="스코로호드 공간" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Funzione_c%C3%A0dl%C3%A0g" title="Funzione càdlàg – Italian" lang="it" hreflang="it" data-title="Funzione càdlàg" 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%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%99%D7%AA_%D7%A7%D7%93%D7%9C%D7%92" title="פונקציית קדלג – Hebrew" lang="he" hreflang="he" data-title="פונקציית קדלג" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/C%C3%A0dl%C3%A0g" title="Càdlàg – Hungarian" lang="hu" hreflang="hu" data-title="Càdlàg" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8F%B3%E9%80%A3%E7%B6%9A%E5%B7%A6%E6%A5%B5%E9%99%90" title="右連続左極限 – Japanese" lang="ja" hreflang="ja" data-title="右連続左極限" 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-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/C%C3%A0dl%C3%A0g" title="Càdlàg – Portuguese" lang="pt" hreflang="pt" data-title="Càdlàg" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%B5%D0%BF%D1%80%D0%B5%D1%80%D1%8B%D0%B2%D0%BD%D0%B0%D1%8F_%D1%81%D0%BF%D1%80%D0%B0%D0%B2%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D1%81_%D0%BB%D0%B5%D0%B2%D0%BE%D1%81%D1%82%D0%BE%D1%80%D0%BE%D0%BD%D0%BD%D0%B8%D0%BC%D0%B8_%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B0%D0%BC%D0%B8" title="Непрерывная справа функция с левосторонними пределами – Russian" lang="ru" hreflang="ru" data-title="Непрерывная справа функция с левосторонними пределами" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/C%C3%A0dl%C3%A0g" title="Càdlàg – Turkish" lang="tr" hreflang="tr" data-title="Càdlàg" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%B5%D0%BF%D0%B5%D1%80%D0%B5%D1%80%D0%B2%D0%BD%D0%B0_%D1%81%D0%BF%D1%80%D0%B0%D0%B2%D0%B0_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%8F_%D0%B7_%D0%BB%D1%96%D0%B2%D0%BE%D1%81%D1%82%D0%BE%D1%80%D0%BE%D0%BD%D0%BD%D1%96%D0%BC%D0%B8_%D0%B3%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D1%8F%D0%BC%D0%B8" title="Неперервна справа функція з лівосторонніми границями – Ukrainian" lang="uk" hreflang="uk" data-title="Неперервна справа функція з лівосторонніми границями" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%C3%A0dl%C3%A0g" title="Càdlàg – Vietnamese" lang="vi" hreflang="vi" data-title="Càdlàg" data-language-autonym="Tiếng Việt" 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Right continuous function with left limits</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>càdlàg</b> (<a href="/wiki/French_language" title="French language">French</a>: <i lang="fr">continue à droite, limite à gauche</i>), <b>RCLL</b> ("right continuous with left limits"), or <b>corlol</b> ("continuous on (the) right, limit on (the) left") function is a function defined on the <a href="/wiki/Real_number" title="Real number">real numbers</a> (or a <a href="/wiki/Subset" title="Subset">subset</a> of them) that is everywhere <a href="/wiki/Right-continuous" class="mw-redirect" title="Right-continuous">right-continuous</a> and has left <a href="/wiki/Limit_of_a_function" title="Limit of a function">limits</a> everywhere. Càdlàg functions are important in the study of <a href="/wiki/Stochastic_processes" class="mw-redirect" title="Stochastic processes">stochastic processes</a> that admit (or even require) jumps, unlike <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a>, which has continuous sample paths. The collection of càdlàg functions on a given <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> is known as <b>Skorokhod space</b>. </p><p>Two related terms are <b>càglàd</b>, standing for "<span title="French-language text"><i lang="fr">continue à gauche, limite à droite</i></span>", the left-right reversal of càdlàg, and <b>càllàl</b> for "<span title="French-language text"><i lang="fr">continue à l'un, limite à l’autre</i></span>" (continuous on one side, limit on the other side), for a function which at each point of the domain is either càdlàg or càglàd. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A0dl%C3%A0g&action=edit&section=1" title="Edit section: Definition"><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:Discrete_probability_distribution_illustration.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Discrete_probability_distribution_illustration.png/220px-Discrete_probability_distribution_illustration.png" decoding="async" width="220" height="248" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Discrete_probability_distribution_illustration.png/330px-Discrete_probability_distribution_illustration.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Discrete_probability_distribution_illustration.png/440px-Discrete_probability_distribution_illustration.png 2x" data-file-width="1806" data-file-height="2033" /></a><figcaption><a href="/wiki/Cumulative_distribution_functions" class="mw-redirect" title="Cumulative distribution functions">Cumulative distribution functions</a> are examples of càdlàg functions.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Discrete_probability_distribution_with_a_countable_set_of_discontinuities.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Discrete_probability_distribution_with_a_countable_set_of_discontinuities.svg/220px-Discrete_probability_distribution_with_a_countable_set_of_discontinuities.svg.png" decoding="async" width="220" height="77" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Discrete_probability_distribution_with_a_countable_set_of_discontinuities.svg/330px-Discrete_probability_distribution_with_a_countable_set_of_discontinuities.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Discrete_probability_distribution_with_a_countable_set_of_discontinuities.svg/440px-Discrete_probability_distribution_with_a_countable_set_of_discontinuities.svg.png 2x" data-file-width="600" data-file-height="210" /></a><figcaption>Example of a cumulative distribution function with a countably infinite set of discontinuities</figcaption></figure> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d78e6f2ddf5baee227ee2a9f164726ba0c23c263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.501ex; height:2.843ex;" alt="{\displaystyle (M,d)}"></span> be a <a href="/wiki/Metric_space" title="Metric space">metric space</a>, and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\subseteq \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\subseteq \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/651e39fa02a0a98bc7f719fb35a883abe09bd8c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.552ex; height:2.343ex;" alt="{\displaystyle E\subseteq \mathbb {R} }"></span>. A function <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:E\to M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:E\to M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/256ed3511e25ca61629b3ffa71176666dde9f6b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.048ex; height:2.509ex;" alt="{\displaystyle f:E\to M}"></span> is called a <b>càdlàg function</b> if, 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 t\in E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e440b636de9fe0ade6ae59c02c2de5b3130d8cf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.456ex; height:2.176ex;" alt="{\displaystyle t\in E}"></span>, </p> <ul><li>the <a href="/wiki/Left_limit" class="mw-redirect" title="Left limit">left limit</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 f(t-):=\lim _{s\to t^{-}}f(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo stretchy="false">→</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t-):=\lim _{s\to t^{-}}f(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3898e8aed1e476c63c153dc9be190888ecd1b673" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.257ex; height:4.343ex;" alt="{\displaystyle f(t-):=\lim _{s\to t^{-}}f(s)}"></span> exists; and</li> <li>the <a href="/wiki/Right_limit" class="mw-redirect" title="Right limit">right limit</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 f(t+):=\lim _{s\to t^{+}}f(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo stretchy="false">→</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t+):=\lim _{s\to t^{+}}f(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81444d61fd03d134d09442c07642df52c67a44ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.257ex; height:4.343ex;" alt="{\displaystyle f(t+):=\lim _{s\to t^{+}}f(s)}"></span> exists and equals <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(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}"></span>.</li></ul> <p>That is, <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is right-continuous with left limits. </p> <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=C%C3%A0dl%C3%A0g&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>All functions continuous on a subset of the real numbers are càdlàg functions on that subset.</li> <li>As a consequence of their definition, all <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution functions</a> are càdlàg functions. For instance the cumulative at point <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> correspond to the probability of being lower or equal than <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>, 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="{\displaystyle \mathbb {P} [X\leq r]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo>≤</mo> <mi>r</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} [X\leq r]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/598fd4e658eb40b575311aed998150911b1f28b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.841ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} [X\leq r]}"></span>. In other words, the semi-open <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> of concern for a two-tailed distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,r]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\infty ,r]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff39189ed8c79947d01d16778300a9713f5c0c07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.766ex; height:2.843ex;" alt="{\displaystyle (-\infty ,r]}"></span> is right-closed.</li> <li>The right derivative <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_{+}^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{+}^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dea27c39e18b98d2b45d169ccc493890e5cad9f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.65ex; height:2.843ex;" alt="{\displaystyle f_{+}^{\prime }}"></span> of any <a href="/wiki/Convex_function" title="Convex function">convex function</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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> defined on an open interval, is an increasing cadlag function.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Skorokhod_space">Skorokhod space</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A0dl%C3%A0g&action=edit&section=3" title="Edit section: Skorokhod space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The set of all càdlàg functions from <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> 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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is often denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {D} (E:M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">D</mi> </mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo>:</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {D} (E:M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03408bdd375dafe665c3bc2b0e6a432c7985164c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.642ex; height:2.843ex;" alt="{\displaystyle \mathbb {D} (E:M)}"></span> (or simply <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 {D} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">D</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {D} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b932b553742ca27776057f1262527014ebbb46a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {D} }"></span>) and is called <b>Skorokhod space</b> after the <a href="/wiki/NASU_Institute_of_Mathematics" title="NASU Institute of Mathematics">Ukrainian mathematician</a> <a href="/wiki/Anatoliy_Skorokhod" title="Anatoliy Skorokhod">Anatoliy Skorokhod</a>. Skorokhod space can be assigned a <a href="/wiki/Topology" title="Topology">topology</a> that intuitively allows us to "wiggle space and time a bit" (whereas the traditional topology of <a href="/wiki/Uniform_convergence" title="Uniform convergence">uniform convergence</a> only allows us to "wiggle space a bit").<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> For simplicity, take <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=[0,T]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=[0,T]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/854c8a073e60092dd8089b8b3e11d4272ba64d2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10ex; height:2.843ex;" alt="{\displaystyle E=[0,T]}"></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 M=\mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f98351dc4e47f515338238b4eed6be6d46357bb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.437ex; height:2.343ex;" alt="{\displaystyle M=\mathbb {R} ^{n}}"></span> — see Billingsley<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 a more general construction. </p><p>We must first define an analogue of the <a href="/wiki/Modulus_of_continuity" title="Modulus of continuity">modulus of continuity</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 \varpi '_{f}(\delta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ϖ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varpi '_{f}(\delta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dd6cc11360b1ebb555827b44fef0b420ba3387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.919ex; height:3.343ex;" alt="{\displaystyle \varpi '_{f}(\delta )}"></span>. 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 F\subseteq E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>⊆</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\subseteq E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3400f80a7838cc279728b12a6e859ca65e353c6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.615ex; height:2.343ex;" alt="{\displaystyle F\subseteq E}"></span>, set </p> <dl><dd><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 w_{f}(F):=\sup _{s,t\in F}|f(s)-f(t)|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>∈</mo> <mi>F</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{f}(F):=\sup _{s,t\in F}|f(s)-f(t)|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/971536adb2e247f013f2024b4aed421538935296" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:26.872ex; height:4.843ex;" alt="{\displaystyle w_{f}(F):=\sup _{s,t\in F}|f(s)-f(t)|}"></span></dd></dl> <p>and, 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 \delta >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/595d5cea06fdcaf2642caf549eda2cfc537958a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta >0}"></span>, define the <b>càdlàg modulus</b> to be </p> <dl><dd><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 \varpi '_{f}(\delta ):=\inf _{\Pi }\max _{1\leq i\leq k}w_{f}([t_{i-1},t_{i})),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ϖ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo movablelimits="true" form="prefix">inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Π</mi> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>k</mi> </mrow> </munder> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varpi '_{f}(\delta ):=\inf _{\Pi }\max _{1\leq i\leq k}w_{f}([t_{i-1},t_{i})),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0d29f0cd91045936dc4a167d606a52d3b6d1dcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.113ex; height:4.343ex;" alt="{\displaystyle \varpi '_{f}(\delta ):=\inf _{\Pi }\max _{1\leq i\leq k}w_{f}([t_{i-1},t_{i})),}"></span></dd></dl> <p>where the <a href="/wiki/Infimum" class="mw-redirect" title="Infimum">infimum</a> runs over all partitions <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 \Pi =\{0=t_{0}<t_{1}<\dots <t_{k}=T\},\;k\in E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Π</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mi>T</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mi>k</mi> <mo>∈</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi =\{0=t_{0}<t_{1}<\dots <t_{k}=T\},\;k\in E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/090f5937f90ff0933f28f1b606f593e1ea1fc4e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.403ex; height:2.843ex;" alt="{\displaystyle \Pi =\{0=t_{0}<t_{1}<\dots <t_{k}=T\},\;k\in E}"></span>, with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \min _{i}(t_{i}-t_{i+1})>\delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>></mo> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \min _{i}(t_{i}-t_{i+1})>\delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/983cb63af72c19c0c1cecb1af4c5861a6c4df1eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.051ex; height:4.009ex;" alt="{\displaystyle \min _{i}(t_{i}-t_{i+1})>\delta }"></span>. This definition makes sense for non-càdlàg <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> (just as the usual modulus of continuity makes sense for discontinuous functions). <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> is càdlàg <a href="/wiki/If_and_only_if" title="If and only if">if and only if</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 \lim _{\delta \to 0}\varpi '_{f}(\delta )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <msubsup> <mi>ϖ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{\delta \to 0}\varpi '_{f}(\delta )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83bb1edf0141c4568300ac2163c0a8e36813095a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.796ex; height:4.176ex;" alt="{\displaystyle \lim _{\delta \to 0}\varpi '_{f}(\delta )=0}"></span>. </p><p>Now let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }"></span> denote the set of all <a href="/wiki/Strictly_increasing" class="mw-redirect" title="Strictly increasing">strictly increasing</a>, continuous <a href="/wiki/Bijection" title="Bijection">bijections</a> from <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> to itself (these are "wiggles in time"). Let </p> <dl><dd><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\|:=\sup _{t\in E}|f(t)|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo fence="false" stretchy="false">‖</mo> <mo>:=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>∈</mo> <mi>E</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|:=\sup _{t\in E}|f(t)|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21f45b3e8ea51a7a8d1aa3f28a937830309a1773" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.458ex; height:4.509ex;" alt="{\displaystyle \|f\|:=\sup _{t\in E}|f(t)|}"></span></dd></dl> <p>denote the uniform norm on functions 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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span>. Define the <b>Skorokhod metric</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span> 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 \mathbb {D} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">D</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {D} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b932b553742ca27776057f1262527014ebbb46a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {D} }"></span> by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (f,g):=\inf _{\lambda \in \Lambda }\max\{\|\lambda -I\|,\|f-g\circ \lambda \|\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo movablelimits="true" form="prefix">inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>∈</mo> <mi mathvariant="normal">Λ</mi> </mrow> </munder> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">‖</mo> <mi>λ</mi> <mo>−</mo> <mi>I</mi> <mo fence="false" stretchy="false">‖</mo> <mo>,</mo> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo>−</mo> <mi>g</mi> <mo>∘</mo> <mi>λ</mi> <mo fence="false" stretchy="false">‖</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (f,g):=\inf _{\lambda \in \Lambda }\max\{\|\lambda -I\|,\|f-g\circ \lambda \|\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb6a4d1f4bbfb14caa3484fe09fa17fac32c5d31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:41.029ex; height:4.176ex;" alt="{\displaystyle \sigma (f,g):=\inf _{\lambda \in \Lambda }\max\{\|\lambda -I\|,\|f-g\circ \lambda \|\},}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I:E\to E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">→</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I:E\to E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2adceb8ece135eabd75d9dbb5a5b11b63767711f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.274ex; height:2.176ex;" alt="{\displaystyle I:E\to E}"></span> is the identity function. In terms of the "wiggle" intuition, <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 \|\lambda -I\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>λ</mi> <mo>−</mo> <mi>I</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\lambda -I\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e411b673d9d96ff05f35e906c8ff079bae9abbb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.692ex; height:2.843ex;" alt="{\displaystyle \|\lambda -I\|}"></span> measures the size of the "wiggle in time", 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 \|f-g\circ \lambda \|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo>−</mo> <mi>g</mi> <mo>∘</mo> <mi>λ</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f-g\circ \lambda \|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e69aed72b497be85771d72d19f503cfd59ead871" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.11ex; height:2.843ex;" alt="{\displaystyle \|f-g\circ \lambda \|}"></span> measures the size of the "wiggle in space". </p><p>The Skorokhod <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a> is indeed a metric. The topology <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }"></span> generated by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span> is called the <b>Skorokhod topology</b> 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 \mathbb {D} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">D</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {D} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b932b553742ca27776057f1262527014ebbb46a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {D} }"></span>. </p><p>An equivalent metric, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(f,g):=\inf _{\lambda \in \Lambda }(\|\lambda -I\|+\|f-g\circ \lambda \|),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo movablelimits="true" form="prefix">inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>∈</mo> <mi mathvariant="normal">Λ</mi> </mrow> </munder> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">‖</mo> <mi>λ</mi> <mo>−</mo> <mi>I</mi> <mo fence="false" stretchy="false">‖</mo> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo>−</mo> <mi>g</mi> <mo>∘</mo> <mi>λ</mi> <mo fence="false" stretchy="false">‖</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(f,g):=\inf _{\lambda \in \Lambda }(\|\lambda -I\|+\|f-g\circ \lambda \|),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98605f00a60cb5ab711b4729cf3e6bf1ae52c441" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:37.493ex; height:4.176ex;" alt="{\displaystyle d(f,g):=\inf _{\lambda \in \Lambda }(\|\lambda -I\|+\|f-g\circ \lambda \|),}"></span></dd></dl> <p>was introduced independently and utilized in control theory for the analysis of switching systems.<sup id="cite_ref-georgiousmith_3-0" class="reference"><a href="#cite_note-georgiousmith-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Properties_of_Skorokhod_space">Properties of Skorokhod space</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A0dl%C3%A0g&action=edit&section=4" title="Edit section: Properties of Skorokhod space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Generalization_of_the_uniform_topology">Generalization of the uniform topology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A0dl%C3%A0g&action=edit&section=5" title="Edit section: Generalization of the uniform topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> of continuous functions 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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> is a <a href="/wiki/Subspace_topology" title="Subspace topology">subspace</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {D} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">D</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {D} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b932b553742ca27776057f1262527014ebbb46a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {D} }"></span>. The Skorokhod topology relativized 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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> coincides with the uniform topology there. </p> <div class="mw-heading mw-heading3"><h3 id="Completeness">Completeness</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A0dl%C3%A0g&action=edit&section=6" title="Edit section: Completeness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although <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 {D} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">D</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {D} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b932b553742ca27776057f1262527014ebbb46a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {D} }"></span> is not a <a href="/wiki/Complete_space" class="mw-redirect" title="Complete space">complete space</a> with respect to the Skorokhod metric <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span>, there is a <a href="/wiki/Metric_(mathematics)#Equivalence_of_metrics" class="mw-redirect" title="Metric (mathematics)">topologically equivalent metric</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f330c7d4a4ee92e5d43dfe3e23f0de3406ec78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.382ex; height:2.009ex;" alt="{\displaystyle \sigma _{0}}"></span> with respect to which <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 {D} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">D</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {D} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b932b553742ca27776057f1262527014ebbb46a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {D} }"></span> is complete.<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-heading3"><h3 id="Separability">Separability</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A0dl%C3%A0g&action=edit&section=7" title="Edit section: Separability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>With respect to either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f330c7d4a4ee92e5d43dfe3e23f0de3406ec78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.382ex; height:2.009ex;" alt="{\displaystyle \sigma _{0}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {D} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">D</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {D} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b932b553742ca27776057f1262527014ebbb46a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {D} }"></span> is a <a href="/wiki/Separable_space" title="Separable space">separable space</a>. Thus, Skorokhod space is a <a href="/wiki/Polish_space" title="Polish space">Polish space</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Tightness_in_Skorokhod_space">Tightness in Skorokhod space</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A0dl%C3%A0g&action=edit&section=8" title="Edit section: Tightness in Skorokhod space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By an application of the <a href="/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem" title="Arzelà–Ascoli theorem">Arzelà–Ascoli theorem</a>, one can show that a sequence <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 _{n})_{n=1,2,\dots }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mu _{n})_{n=1,2,\dots }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b693382670163b5cb616a613f1cf5017830884cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.411ex; height:3.009ex;" alt="{\displaystyle (\mu _{n})_{n=1,2,\dots }}"></span> of <a href="/wiki/Probability_measure" title="Probability measure">probability measures</a> on Skorokhod space <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 {D} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">D</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {D} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b932b553742ca27776057f1262527014ebbb46a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {D} }"></span> is <a href="/wiki/Tightness_of_measures" title="Tightness of measures">tight</a> if and only if both the following conditions are met: </p> <dl><dd><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 \lim _{a\to \infty }\limsup _{n\to \infty }\mu _{n}{\big (}\{f\in \mathbb {D} \;|\;\|f\|\geq a\}{\big )}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">lim sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>μ</mi> <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> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">D</mi> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thickmathspace" /> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo fence="false" stretchy="false">‖</mo> <mo>≥</mo> <mi>a</mi> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{a\to \infty }\limsup _{n\to \infty }\mu _{n}{\big (}\{f\in \mathbb {D} \;|\;\|f\|\geq a\}{\big )}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ebfb4f82b6c7fcac62f6ae60657af69eaa434c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.698ex; height:4.509ex;" alt="{\displaystyle \lim _{a\to \infty }\limsup _{n\to \infty }\mu _{n}{\big (}\{f\in \mathbb {D} \;|\;\|f\|\geq a\}{\big )}=0,}"></span></dd></dl> <p>and </p> <dl><dd><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 \lim _{\delta \to 0}\limsup _{n\to \infty }\mu _{n}{\big (}\{f\in \mathbb {D} \;|\;\varpi '_{f}(\delta )\geq \varepsilon \}{\big )}=0{\text{ for all }}\varepsilon >0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">lim sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>μ</mi> <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> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">D</mi> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thickmathspace" /> <msubsup> <mi>ϖ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>ε</mi> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>ε</mi> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{\delta \to 0}\limsup _{n\to \infty }\mu _{n}{\big (}\{f\in \mathbb {D} \;|\;\varpi '_{f}(\delta )\geq \varepsilon \}{\big )}=0{\text{ for all }}\varepsilon >0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5a54e76176e46885a37a11fa0ea75bc6c0614c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:53.269ex; height:4.509ex;" alt="{\displaystyle \lim _{\delta \to 0}\limsup _{n\to \infty }\mu _{n}{\big (}\{f\in \mathbb {D} \;|\;\varpi '_{f}(\delta )\geq \varepsilon \}{\big )}=0{\text{ for all }}\varepsilon >0.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Algebraic_and_topological_structure">Algebraic and topological structure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A0dl%C3%A0g&action=edit&section=9" title="Edit section: Algebraic and topological structure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Under the Skorokhod topology and pointwise addition of functions, <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 {D} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">D</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {D} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b932b553742ca27776057f1262527014ebbb46a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {D} }"></span> is not a topological group, as can be seen by the following example: </p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=[0,2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=[0,2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c617d754ab1d0c2f3ef3c3c7e6cfac280cfedf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.784ex; height:2.843ex;" alt="{\displaystyle E=[0,2)}"></span> be a half-open interval and take <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_{n}=\chi _{[1-1/n,2)}\in \mathbb {D} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">D</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{n}=\chi _{[1-1/n,2)}\in \mathbb {D} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6633f113c8be0d0beac9b575afd31a18a8e07103" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.769ex; height:3.009ex;" alt="{\displaystyle f_{n}=\chi _{[1-1/n,2)}\in \mathbb {D} }"></span> to be a sequence of characteristic functions. Despite the fact 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 f_{n}\rightarrow \chi _{[1,2)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{n}\rightarrow \chi _{[1,2)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2238ec71ff7bf2dab20e54f7080fd05170c08ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.858ex; height:3.009ex;" alt="{\displaystyle f_{n}\rightarrow \chi _{[1,2)}}"></span> in the Skorokhod topology, the sequence <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_{n}-\chi _{[1,2)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{n}-\chi _{[1,2)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f79046f3f8bd4c4e036ecb4bd30c5a1ab595592b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.084ex; height:3.009ex;" alt="{\displaystyle f_{n}-\chi _{[1,2)}}"></span> does not converge to 0. </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=C%C3%A0dl%C3%A0g&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Classical_Wiener_space" title="Classical Wiener space">Classical Wiener space</a></li></ul> <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=C%C3%A0dl%C3%A0g&action=edit&section=11" title="Edit section: References"><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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Skorokhod_space">"Skorokhod space - Encyclopedia of Mathematics"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Skorokhod+space+-+Encyclopedia+of+Mathematics&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FSkorokhod_space&rfr_id=info%3Asid%2Fen.wikipedia.org%3AC%C3%A0dl%C3%A0g" 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="CITEREFBillingsley" class="citation book cs1">Billingsley, P. <i>Convergence of Probability Measures</i>. New York: Wiley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Convergence+of+Probability+Measures&rft.place=New+York&rft.pub=Wiley&rft.aulast=Billingsley&rft.aufirst=P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AC%C3%A0dl%C3%A0g" class="Z3988"></span></span> </li> <li id="cite_note-georgiousmith-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-georgiousmith_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorgiou,_T.T._and_Smith,_M.C.2000" class="citation journal cs1">Georgiou, T.T. and Smith, M.C. (2000). "Robustness of a relaxation oscillator". <i>International Journal of Robust and Nonlinear Control</i>. <b>10</b> (11–12): 1005–1024. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F1099-1239%28200009%2F10%2910%3A11%2F12%3C1005%3A%3AAID-RNC536%3E3.0.CO%3B2-Q">10.1002/1099-1239(200009/10)10:11/12<1005::AID-RNC536>3.0.CO;2-Q</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Robust+and+Nonlinear+Control&rft.atitle=Robustness+of+a+relaxation+oscillator&rft.volume=10&rft.issue=11%E2%80%9312&rft.pages=1005-1024&rft.date=2000&rft_id=info%3Adoi%2F10.1002%2F1099-1239%28200009%2F10%2910%3A11%2F12%3C1005%3A%3AAID-RNC536%3E3.0.CO%3B2-Q&rft.au=Georgiou%2C+T.T.+and+Smith%2C+M.C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AC%C3%A0dl%C3%A0g" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBillingsley" class="citation book cs1">Billingsley, P. <i>Convergence of Probability Measures</i>. New York: Wiley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Convergence+of+Probability+Measures&rft.place=New+York&rft.pub=Wiley&rft.aulast=Billingsley&rft.aufirst=P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AC%C3%A0dl%C3%A0g" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%C3%A0dl%C3%A0g&action=edit&section=12" title="Edit section: Further reading"><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 book cs1">Billingsley, Patrick (1995). <i>Probability and Measure</i>. New York, NY: John Wiley & Sons, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-00710-2" title="Special:BookSources/0-471-00710-2"><bdi>0-471-00710-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability+and+Measure&rft.place=New+York%2C+NY&rft.pub=John+Wiley+%26+Sons%2C+Inc.&rft.date=1995&rft.isbn=0-471-00710-2&rft.au=Billingsley%2C+Patrick&rfr_id=info%3Asid%2Fen.wikipedia.org%3AC%C3%A0dl%C3%A0g" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1">Billingsley, Patrick (1999). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/convergenceofpro0000bill"><i>Convergence of Probability Measures</i></a></span>. New York, NY: John Wiley & Sons, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-19745-9" title="Special:BookSources/0-471-19745-9"><bdi>0-471-19745-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Convergence+of+Probability+Measures&rft.place=New+York%2C+NY&rft.pub=John+Wiley+%26+Sons%2C+Inc.&rft.date=1999&rft.isbn=0-471-19745-9&rft.au=Billingsley%2C+Patrick&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fconvergenceofpro0000bill&rfr_id=info%3Asid%2Fen.wikipedia.org%3AC%C3%A0dl%C3%A0g" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Càdlàg&oldid=1255530943">https://en.wikipedia.org/w/index.php?title=Càdlàg&oldid=1255530943</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Real_analysis" title="Category:Real analysis">Real analysis</a></li><li><a href="/wiki/Category:Stochastic_processes" title="Category:Stochastic processes">Stochastic processes</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">CS1 maint: multiple names: authors list</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_containing_French-language_text" title="Category:Articles containing French-language text">Articles containing French-language text</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 5 November 2024, at 11:46<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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