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id="toc-Definición-sublist" class="vector-toc-list"> <li id="toc-Definición_discreta" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definición_discreta"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Definición discreta</span> </div> </a> <ul id="toc-Definición_discreta-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definición_usando_filtraciones" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definición_usando_filtraciones"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Definición usando filtraciones</span> </div> </a> <ul id="toc-Definición_usando_filtraciones-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bibliografía" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografía"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Bibliografía</span> </div> </a> <ul id="toc-Bibliografía-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Enlaces_externos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Enlaces_externos"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Enlaces externos</span> </div> </a> <ul id="toc-Enlaces_externos-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contenidos" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Cambiar a la tabla de contenidos" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Cambiar a la tabla de contenidos</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Tiempo de parada</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="Ir a un artículo en otro idioma. Disponible en 12 idiomas" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-12" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">12 idiomas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Stoppzeit" title="Stoppzeit (alemán)" lang="de" hreflang="de" data-title="Stoppzeit" data-language-autonym="Deutsch" data-language-local-name="alemán" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Stopping_time" title="Stopping time (inglés)" lang="en" hreflang="en" data-title="Stopping time" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B2%D9%85%D8%A7%D9%86_%D8%AA%D9%88%D9%82%D9%81" title="زمان توقف (persa)" lang="fa" hreflang="fa" data-title="زمان توقف" data-language-autonym="فارسی" data-language-local-name="persa" 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/Temps_d%27arr%C3%AAt" title="Temps d'arrêt (francés)" lang="fr" hreflang="fr" data-title="Temps d'arrêt" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%96%D7%9E%D7%9F_%D7%A2%D7%A6%D7%99%D7%A8%D7%94" title="זמן עצירה (hebreo)" lang="he" hreflang="he" data-title="זמן עצירה" data-language-autonym="עברית" data-language-local-name="hebreo" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Tempo_di_arresto" title="Tempo di arresto (italiano)" lang="it" hreflang="it" data-title="Tempo di arresto" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%95%EC%A7%80_%EC%8B%9C%EA%B0%84" title="정지 시간 (coreano)" lang="ko" hreflang="ko" data-title="정지 시간" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Stoptijd" title="Stoptijd (neerlandés)" lang="nl" hreflang="nl" data-title="Stoptijd" data-language-autonym="Nederlands" data-language-local-name="neerlandés" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Moment_zatrzymania" title="Moment zatrzymania (polaco)" lang="pl" hreflang="pl" data-title="Moment zatrzymania" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%80%D0%BA%D0%BE%D0%B2%D1%81%D0%BA%D0%B8%D0%B9_%D0%BC%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%B2%D1%80%D0%B5%D0%BC%D0%B5%D0%BD%D0%B8" title="Марковский момент времени (ruso)" lang="ru" hreflang="ru" data-title="Марковский момент времени" data-language-autonym="Русский" data-language-local-name="ruso" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%80%D0%BA%D0%BE%D0%B2%D1%81%D1%8C%D0%BA%D0%B8%D0%B9_%D0%BC%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D1%87%D0%B0%D1%81%D1%83" title="Марковський момент часу (ucraniano)" lang="uk" hreflang="uk" data-title="Марковський момент часу" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%81%9C%E6%97%B6" title="停时 (chino)" lang="zh" hreflang="zh" data-title="停时" data-language-autonym="中文" data-language-local-name="chino" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1288707#sitelinks-wikipedia" title="Editar enlaces interlingüísticos" class="wbc-editpage">Editar enlaces</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espacios de nombres"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Tiempo_de_parada" title="Ver la página de contenido [c]" accesskey="c"><span>Artículo</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Discusi%C3%B3n:Tiempo_de_parada&action=edit&redlink=1" rel="discussion" class="new" title="Discusión acerca de la página (aún no redactado) [t]" accesskey="t"><span>Discusión</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Cambiar variante de idioma" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">español</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Vistas"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Tiempo_de_parada"><span>Leer</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Tiempo_de_parada&action=edit" title="Editar esta página [e]" accesskey="e"><span>Editar</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Tiempo_de_parada&action=history" title="Versiones anteriores de esta página [h]" accesskey="h"><span>Ver historial</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Página de herramientas"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Herramientas" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Herramientas</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Herramientas</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mover a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">ocultar</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Más opciones" > <div class="vector-menu-heading"> Acciones </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Tiempo_de_parada"><span>Leer</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Tiempo_de_parada&action=edit" title="Editar esta página [e]" accesskey="e"><span>Editar</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Tiempo_de_parada&action=history"><span>Ver historial</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Especial:LoQueEnlazaAqu%C3%AD/Tiempo_de_parada" title="Lista de todas las páginas de la wiki que enlazan aquí [j]" accesskey="j"><span>Lo que enlaza aquí</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:CambiosEnEnlazadas/Tiempo_de_parada" rel="nofollow" title="Cambios recientes en las páginas que enlazan con esta [k]" accesskey="k"><span>Cambios en enlazadas</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=es" title="Subir archivos [u]" accesskey="u"><span>Subir archivo</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Especial:P%C3%A1ginasEspeciales" title="Lista de todas las páginas especiales [q]" accesskey="q"><span>Páginas especiales</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Tiempo_de_parada&oldid=155207248" title="Enlace permanente a esta versión de la página"><span>Enlace permanente</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Tiempo_de_parada&action=info" title="Más información sobre esta página"><span>Información de la página</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Especial:Citar&page=Tiempo_de_parada&id=155207248&wpFormIdentifier=titleform" title="Información sobre cómo citar esta página"><span>Citar esta página</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Especial:Acortador_de_URL&url=https%3A%2F%2Fes.wikipedia.org%2Fwiki%2FTiempo_de_parada"><span>Obtener URL acortado</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Especial:QrCode&url=https%3A%2F%2Fes.wikipedia.org%2Fwiki%2FTiempo_de_parada"><span>Descargar código QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Imprimir/exportar </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Especial:Libro&bookcmd=book_creator&referer=Tiempo+de+parada"><span>Crear un libro</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Especial:DownloadAsPdf&page=Tiempo_de_parada&action=show-download-screen"><span>Descargar como PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Tiempo_de_parada&printable=yes" title="Versión imprimible de esta página [p]" accesskey="p"><span>Versión para imprimir</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> En otros proyectos </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1288707" title="Enlace al elemento conectado del repositorio de datos [g]" accesskey="g"><span>Elemento de Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Página de herramientas"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Apariencia"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Apariencia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mover a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ocultar</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" 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">De Wikipedia, la enciclopedia libre</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="es" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Archivo:HittingTimes1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/HittingTimes1.png/220px-HittingTimes1.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/HittingTimes1.png/330px-HittingTimes1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/HittingTimes1.png/440px-HittingTimes1.png 2x" data-file-width="1200" data-file-height="900" /></a><figcaption>Ejemplo de tiempo de parada en un <a href="/wiki/Movimiento_browniano" title="Movimiento browniano">Movimiento browniano</a>.</figcaption></figure> <p>En <a href="/wiki/Teor%C3%ADa_de_la_probabilidad" title="Teoría de la probabilidad">teoría de la probabilidad</a>, en particular en el estudio de los <a href="/wiki/Procesos_estoc%C3%A1sticos" class="mw-redirect" title="Procesos estocásticos">procesos estocásticos</a>, el <b>tiempo de parada</b> (también conocido como <b>tiempo de Markov</b>) es un tipo específico de «tiempo aleatorio». </p><p>La teoría del tiempo de parada puede ser analizada en <a href="/wiki/Probabilidad" title="Probabilidad">probabilidad</a> o en <a href="/wiki/Estad%C3%ADstica" title="Estadística">estadística</a>, notablemente a través del <a href="/wiki/Teorema_de_la_parada_opcional" title="Teorema de la parada opcional">teorema de la parada opcional</a>. Además, los tiempos de parada son frecuentemente aplicados en pruebas <a href="/wiki/Matem%C3%A1ticas" title="Matemáticas">matemáticas</a> para «controlar el continuum de tiempo», como Chung lo expresó en su libro en 1982. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definición"><span id="Definici.C3.B3n"></span>Definición</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tiempo_de_parada&action=edit&section=1" title="Editar sección: Definición"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Definición_discreta"><span id="Definici.C3.B3n_discreta"></span>Definición discreta</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tiempo_de_parada&action=edit&section=2" title="Editar sección: Definición discreta"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se dice que una variable aleatoria <span class="mwe-math-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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> es un tiempo de parada respecto de las variables aleatorias <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X_{0},X_{1},...,X_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X_{0},X_{1},...,X_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0709a3999c785fe00c6fbb86027e6bdf54713c89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.628ex; height:2.843ex;" alt="{\displaystyle \{X_{0},X_{1},...,X_{n}\}}"></span> si basados en la información proporcionada por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X_{0},X_{1},...,X_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X_{0},X_{1},...,X_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0709a3999c785fe00c6fbb86027e6bdf54713c89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.628ex; height:2.843ex;" alt="{\displaystyle \{X_{0},X_{1},...,X_{n}\}}"></span> podemos determinar si el evento <span class="mwe-math-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=n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>T</mi> <mo>=</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{T=n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4af4f843181bc6bbf93b09c0b81db15680957471" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.454ex; height:2.843ex;" alt="{\displaystyle \{T=n\}}"></span> ha ocurrido. El evento <span class="mwe-math-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=n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>T</mi> <mo>=</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{T=n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4af4f843181bc6bbf93b09c0b81db15680957471" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.454ex; height:2.843ex;" alt="{\displaystyle \{T=n\}}"></span> no es el único que queda determinado por la información proporcionada por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X_{0},X_{1},...,X_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X_{0},X_{1},...,X_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0709a3999c785fe00c6fbb86027e6bdf54713c89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.628ex; height:2.843ex;" alt="{\displaystyle \{X_{0},X_{1},...,X_{n}\}}"></span>. De hecho, existen otros eventos como <span class="mwe-math-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\leq n\},\{T\geq n\},\{T<n\}{\text{ y }}\{T>n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>T</mi> <mo>≤</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mi>T</mi> <mo>≥</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mi>T</mi> <mo><</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> y </mtext> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>T</mi> <mo>></mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{T\leq n\},\{T\geq n\},\{T<n\}{\text{ y }}\{T>n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31fe05e357051b8bb9cf4ed4fc07303c62628692" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.274ex; height:2.843ex;" alt="{\displaystyle \{T\leq n\},\{T\geq n\},\{T<n\}{\text{ y }}\{T>n\}}"></span>. </p><p>Otra definición basada en procesos estocásticos es la siguiente: sea <span class="mwe-math-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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> una variable aleatoria definida sobre <span class="mwe-math-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 {N} _{0}\cup \{\infty \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} _{0}\cup \{\infty \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34bf96ac3628e620f678732036d342ec03d1b2bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.964ex; height:2.843ex;" alt="{\displaystyle \mathbb {N} _{0}\cup \{\infty \}}"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{k})_{k\geq 0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{k})_{k\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/134392cd0f69f7cdecbcc4667187eb035301981a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.012ex; height:2.843ex;" alt="{\displaystyle (X_{k})_{k\geq 0}}"></span> un proceso estocástico. Si para todo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 0}"></span> todos los eventos <span class="mwe-math-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\leq n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>T</mi> <mo>≤</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{T\leq n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e33657d5b523e5d331dfdc5a5b395b8e5cd8f156" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.454ex; height:2.843ex;" alt="{\displaystyle \{T\leq n\}}"></span> están determinados totalmente por los elementos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{0},X_{1},...,X_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{0},X_{1},...,X_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33e3d1fbb2fdc9bbb949594928d9c564965af1b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.113ex; height:2.843ex;" alt="{\displaystyle (X_{0},X_{1},...,X_{n})}"></span> se dice que <span class="mwe-math-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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> es un tiempo de parada. También recibe el nombre de tiempo de Markov. </p> <div class="mw-heading mw-heading3"><h3 id="Definición_usando_filtraciones"><span id="Definici.C3.B3n_usando_filtraciones"></span>Definición usando filtraciones</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tiempo_de_parada&action=edit&section=3" title="Editar sección: Definición usando filtraciones"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sea <span class="mwe-math-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 (\Omega ,{\mathcal {F}},\mathbb {F} =({\mathcal {F}}_{t})_{t\in T},P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mrow> </msub> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {F} =({\mathcal {F}}_{t})_{t\in T},P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e871ee75d29a838560573a594ba1fd7bbad2db0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.165ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {F} =({\mathcal {F}}_{t})_{t\in T},P)}"></span> un <a href="/wiki/Espacio_de_probabilidad" title="Espacio de probabilidad">espacio de probabilidad</a> filtrado, donde <span class="mwe-math-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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> es un conjunto de índices, a menudo <span class="mwe-math-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=\mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=\mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4857fb1a69a6e1220b5e2e673d39aa55a5821a8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.413ex; height:2.176ex;" alt="{\displaystyle T=\mathbb {N} }"></span> o <span class="mwe-math-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=[0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=[0,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2d5792c68149c5c2f3e434e8c119a702ced9846" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.806ex; height:2.843ex;" alt="{\displaystyle T=[0,\infty )}"></span>. Un <b>tiempo de parada</b> es una variable aleatoria <span class="mwe-math-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 \tau :\Omega \rightarrow T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>:</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">→</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau :\Omega \rightarrow T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9cb5af94133dd6659d3173a818d0f89bf655d71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.068ex; height:2.176ex;" alt="{\displaystyle \tau :\Omega \rightarrow T}"></span> tal que <span class="mwe-math-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 \{\tau =t\}\in {\mathcal {F}}_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>τ</mi> <mo>=</mo> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\tau =t\}\in {\mathcal {F}}_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae99ccbd261f11cd27fe015aea087401d7d03a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.803ex; height:2.843ex;" alt="{\displaystyle \{\tau =t\}\in {\mathcal {F}}_{t}}"></span> para todo <span class="mwe-math-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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4fe93f70df3818ecca67c2ca44f087483951856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.317ex; height:2.176ex;" alt="{\displaystyle t\in T}"></span>. </p><p>El tiempo de parada tiene lugar en la <a href="/wiki/Teor%C3%ADa_de_la_decisi%C3%B3n" title="Teoría de la decisión">teoría de la decisión</a>, en la cual una <b>regla de parada</b> se caracteriza como un mecanismo que sirve para decidir si continuar o detener un proceso sobre la base de la posición <a href="/wiki/Presente_(tiempo)" title="Presente (tiempo)">presente</a> y de eventos pasados, y que <a href="/wiki/Casi_seguramente" title="Casi seguramente">casi seguramente</a> conduce a una decisión para detener en algún momento de tiempo. </p> <div class="mw-heading mw-heading2"><h2 id="Bibliografía"><span id="Bibliograf.C3.ADa"></span>Bibliografía</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tiempo_de_parada&action=edit&section=4" title="Editar sección: Bibliografía"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span id="CITAREFChung1982" class="citation libro">Chung, Kai Lai (1982). <i>Lectures from Markov processes to Brownian motion</i>. Grundlehren der Mathematischen Wissenschaften No. 249. Nueva York: Springer-Verlag. <small><a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Especial:FuentesDeLibros/0-387-90618-5" title="Especial:FuentesDeLibros/0-387-90618-5">0-387-90618-5</a></small>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3ATiempo+de+parada&rft.au=Chung%2C+Kai+Lai&rft.aufirst=Kai+Lai&rft.aulast=Chung&rft.btitle=Lectures+from+Markov+processes+to+Brownian+motion&rft.date=1982&rft.genre=book&rft.isbn=0-387-90618-5&rft.place=Nueva+York&rft.pub=Springer-Verlag&rft.series=Grundlehren+der+Mathematischen+Wissenschaften+No.+249&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFFischer2013" class="citation publicación">Fischer, Tom (2013). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016/j.spl.2012.09.024">«On simple representations of stopping times and stopping time sigma-algebras»</a>. <i>Statistics and Probability Letters</i> <b>83</b> (1): 345-349. <small><a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016%2Fj.spl.2012.09.024">10.1016/j.spl.2012.09.024</a></small>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3ATiempo+de+parada&rft.atitle=On+simple+representations+of+stopping+times+and+stopping+time+sigma-algebras&rft.au=Fischer%2C+Tom&rft.aufirst=Tom&rft.aulast=Fischer&rft.date=2013&rft.genre=article&rft.issue=1&rft.jtitle=Statistics+and+Probability+Letters&rft.pages=345-349&rft.volume=83&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1016%2Fj.spl.2012.09.024&rft_id=info%3Adoi%2F10.1016%2Fj.spl.2012.09.024&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFRevuz,_Daniel_and_Yor,_Marc1999" class="citation libro">Revuz, Daniel and Yor, Marc (1999). <i>Continuous martingales and Brownian motion</i>. Grundlehren der Mathematischen Wissenschaften No. 293 (Tercera edición edición). Berlín: Springer-Verlag. <small><a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Especial:FuentesDeLibros/3-540-64325-7" title="Especial:FuentesDeLibros/3-540-64325-7">3-540-64325-7</a></small>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3ATiempo+de+parada&rft.au=Revuz%2C+Daniel+and+Yor%2C+Marc&rft.aulast=Revuz%2C+Daniel+and+Yor%2C+Marc&rft.btitle=Continuous+martingales+and+Brownian+motion&rft.date=1999&rft.edition=Tercera+edici%C3%B3n&rft.genre=book&rft.isbn=3-540-64325-7&rft.place=Berl%C3%ADn&rft.pub=Springer-Verlag&rft.series=Grundlehren+der+Mathematischen+Wissenschaften+No.+293&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFH._Vincent_Poor_and_Olympia_Hadjiliadis2008" class="citation libro">H. Vincent Poor and Olympia Hadjiliadis (2008). <i>Quickest Detection</i> (Primera edición edición). Cambridge: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <small><a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Especial:FuentesDeLibros/978-0-521-62104-5" title="Especial:FuentesDeLibros/978-0-521-62104-5">978-0-521-62104-5</a></small>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3ATiempo+de+parada&rft.au=H.+Vincent+Poor+and+Olympia+Hadjiliadis&rft.aulast=H.+Vincent+Poor+and+Olympia+Hadjiliadis&rft.btitle=Quickest+Detection&rft.date=2008&rft.edition=Primera+edici%C3%B3n&rft.genre=book&rft.isbn=978-0-521-62104-5&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><span id="CITAREFProtter2005" class="citation libro">Protter, Philip E. (2005). <i>Stochastic integration and differential equations</i>. Stochastic Modelling and Applied Probability No. 21 (Segunda edición (versión 2.1, corregida tercera impresión) edición). Berlín: Springer-Verlag. <small><a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Especial:FuentesDeLibros/3-540-00313-4" title="Especial:FuentesDeLibros/3-540-00313-4">3-540-00313-4</a></small>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3ATiempo+de+parada&rft.au=Protter%2C+Philip+E.&rft.aufirst=Philip+E.&rft.aulast=Protter&rft.btitle=Stochastic+integration+and+differential+equations&rft.date=2005&rft.edition=Segunda+edici%C3%B3n+%28versi%C3%B3n+2.1%2C+corregida+tercera+impresi%C3%B3n%29&rft.genre=book&rft.isbn=3-540-00313-4&rft.place=Berl%C3%ADn&rft.pub=Springer-Verlag&rft.series=Stochastic+Modelling+and+Applied+Probability+No.+21&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Enlaces_externos">Enlaces externos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tiempo_de_parada&action=edit&section=5" title="Editar sección: Enlaces externos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><div class="plainlinks" style="display:inline">Esta obra contiene una traducción derivada de «<a href="https://en.wikipedia.org/wiki/Stopping_time" class="extiw" title="en:Stopping time">Stopping time</a>» de Wikipedia en inglés, publicada por <a class="external text" href="https://en.wikipedia.org/wiki/Stopping_time?action=history">sus editores</a> bajo la <a href="/wiki/Wikipedia:Texto_de_la_Licencia_de_documentaci%C3%B3n_libre_de_GNU" title="Wikipedia:Texto de la Licencia de documentación libre de GNU">Licencia de documentación libre de GNU</a> y la <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.es">Licencia Creative Commons Atribución-CompartirIgual 4.0 Internacional</a>.</div></li></ul> <style data-mw-deduplicate="TemplateStyles:r161257576">.mw-parser-output .mw-authority-control{margin-top:1.5em}.mw-parser-output .mw-authority-control .navbox table{margin:0}.mw-parser-output .mw-authority-control .navbox hr:last-child{display:none}.mw-parser-output .mw-authority-control .navbox+.mw-mf-linked-projects{display:none}.mw-parser-output .mw-authority-control .mw-mf-linked-projects{display:flex;padding:0.5em;border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-neutral,#eaecf0);color:var(--color-base,#202122)}.mw-parser-output .mw-authority-control .mw-mf-linked-projects ul li{margin-bottom:0}.mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .mw-authority-control .navbox-list{border-color:#f8f9fa}.mw-parser-output .mw-authority-control .navbox th{background-color:#eeeeff}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .mw-mf-linked-projects{border:1px solid var(--border-color-base,#72777d);background-color:var(--background-color-neutral,#27292d);color:var(--color-base,#eaecf0)}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral-subtle,#202122)!important}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox-list{border-color:#202122!important}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox th{background-color:#27292d!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .mw-mf-linked-projects{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral,#27292d)!important;color:var(--color-base,#eaecf0)!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral-subtle,#202122)!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox-list{border-color:#202122!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox th{background-color:#27292d!important}}</style><div class="mw-authority-control"><div role="navigation" class="navbox" aria-label="Navbox" style="width: inherit;padding:3px"><table class="hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width: 12%; 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