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href="/w/index.php?title=%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%A1%D0%B2%D0%B5%D0%B6%D0%B8%D0%B5_%D0%BF%D1%80%D0%B0%D0%B2%D0%BA%D0%B8&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin-vector-legacy mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Марковский_момент_времени rootpage-Марковский_момент_времени skin-vector action-view"><div id="mw-page-base" class="noprint"></div> <div id="mw-head-base" class="noprint"></div> <div id="content" class="mw-body" role="main"> <a id="top"></a> <div id="siteNotice"></div> <div class="mw-indicators"> </div> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Марковский момент времени</span></h1> <div id="bodyContent" class="vector-body"> <div id="siteSub" class="noprint">Материал из Википедии — свободной энциклопедии</div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="contentSub2"></div> <div id="jump-to-nav"></div> <a class="mw-jump-link" href="#mw-head">Перейти к навигации</a> <a class="mw-jump-link" href="#searchInput">Перейти к поиску</a> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ru" dir="ltr"><p><b>Марковский момент <a href="/wiki/%D0%92%D1%80%D0%B5%D0%BC%D1%8F" title="Время">времени</a></b> (в <a href="/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D1%81%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D1%8B%D1%85_%D0%BF%D1%80%D0%BE%D1%86%D0%B5%D1%81%D1%81%D0%BE%D0%B2" class="mw-redirect" title="Теория случайных процессов">теории случайных процессов</a>) — это случайная величина, не зависящая от <a href="/wiki/%D0%91%D1%83%D0%B4%D1%83%D1%89%D0%B5%D0%B5" title="Будущее">будущего</a> рассматриваемого <a href="/wiki/%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D1%8B%D0%B9_%D0%BF%D1%80%D0%BE%D1%86%D0%B5%D1%81%D1%81" title="Случайный процесс">случайного процесса</a>. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="ru" dir="ltr"><h2 id="mw-toc-heading">Содержание</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Дискретный_случай"><span class="tocnumber">1</span> <span class="toctext">Дискретный случай</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Пример"><span class="tocnumber">1.1</span> <span class="toctext">Пример</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-3"><a href="#Общий_случай"><span class="tocnumber">2</span> <span class="toctext">Общий случай</span></a> <ul> <li class="toclevel-2 tocsection-4"><a href="#Свойства"><span class="tocnumber">2.1</span> <span class="toctext">Свойства</span></a></li> <li class="toclevel-2 tocsection-5"><a href="#Пример_2"><span class="tocnumber">2.2</span> <span class="toctext">Пример</span></a></li> </ul> </li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Дискретный_случай"><span id=".D0.94.D0.B8.D1.81.D0.BA.D1.80.D0.B5.D1.82.D0.BD.D1.8B.D0.B9_.D1.81.D0.BB.D1.83.D1.87.D0.B0.D0.B9"></span>Дискретный случай</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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&veaction=edit&section=1" title="Редактировать раздел «Дискретный случай»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%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&action=edit&section=1" title="Редактировать код раздела «Дискретный случай»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Пусть дана последовательность случайных величин <span class="mwe-math-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 \{Y_{n}\}_{n\geq 0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{Y_{n}\}_{n\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0007244167820a85c28e873c11f3a2789989253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.213ex; height:2.843ex;" alt="{\displaystyle \{Y_{n}\}_{n\geq 0}}"></span>. Тогда <a href="/wiki/%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%B0%D1%8F_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0" title="Случайная величина">случайная величина</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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> называется марковским моментом (времени), если для любого <span class="mwe-math-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> событие <span class="mwe-math-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 \leq n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>τ</mi> <mo>≤</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\tau \leq n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f50e1f00a3252d457582ee1c0a43431d6ffd22fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.02ex; height:2.843ex;" alt="{\displaystyle \{\tau \leq n\}}"></span> зависит только от случайных величин <span class="mwe-math-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 Y_{0},\ldots ,Y_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{0},\ldots ,Y_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56628c7f905629a26aea9553a1dae150242c1622" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.152ex; height:2.509ex;" alt="{\displaystyle Y_{0},\ldots ,Y_{n}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Пример"><span id=".D0.9F.D1.80.D0.B8.D0.BC.D0.B5.D1.80"></span>Пример</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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&veaction=edit&section=2" title="Редактировать раздел «Пример»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%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&action=edit&section=2" title="Редактировать код раздела «Пример»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Пусть <span class="mwe-math-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 \{Y_{n}\}_{n\geq 0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{Y_{n}\}_{n\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0007244167820a85c28e873c11f3a2789989253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.213ex; height:2.843ex;" alt="{\displaystyle \{Y_{n}\}_{n\geq 0}}"></span> — последовательность независимых нормальных случайных величин. Пусть <span class="mwe-math-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 L\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0913a2035a5674d95af33a6c3fb41d2e146dc273" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.102ex; height:2.176ex;" alt="{\displaystyle L\in \mathbb {R} }"></span>, и </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 \tau =\inf\{n\geq 0\mid Y_{n}\geq L\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">inf</mo> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo>≥</mo> <mn>0</mn> <mo>∣</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≥</mo> <mi>L</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =\inf\{n\geq 0\mid Y_{n}\geq L\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fd702bb1775e950aacafda4c561bed1e6443b05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.12ex; height:2.843ex;" alt="{\displaystyle \tau =\inf\{n\geq 0\mid Y_{n}\geq L\}}"></span></dd></dl> <p>— момент первого достижения процессом <span class="mwe-math-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 \{Y_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>Y</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 \{Y_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58a38bdd3c7cbd3a4b6362ec1a5e69a7d5766872" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.894ex; height:2.843ex;" alt="{\displaystyle \{Y_{n}\}}"></span> уровня <span class="mwe-math-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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span>. Тогда <span class="mwe-math-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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> — марковский момент, ибо <span class="mwe-math-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 \leq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>≤</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau \leq n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30aab65e2457a20ad39d07de22f0694dae3f6aeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.695ex; height:2.176ex;" alt="{\displaystyle \tau \leq n}"></span> тогда и только тогда, когда существует <span class="mwe-math-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\in \mathbb {N} ,\;0\leq i\leq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mn>0</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in \mathbb {N} ,\;0\leq i\leq n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0cac30d11677c150e79caf35e98df4a8f2f1709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.557ex; height:2.509ex;" alt="{\displaystyle i\in \mathbb {N} ,\;0\leq i\leq n}"></span> такое, что <span class="mwe-math-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 Y_{i}\geq L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≥</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{i}\geq L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02522dcdbd4276b6ab4aa3bfbdc687ccac528964" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.832ex; height:2.509ex;" alt="{\displaystyle Y_{i}\geq L}"></span>. Таким образом событие <span class="mwe-math-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 \leq n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>τ</mi> <mo>≤</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\tau \leq n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f50e1f00a3252d457582ee1c0a43431d6ffd22fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.02ex; height:2.843ex;" alt="{\displaystyle \{\tau \leq n\}}"></span> зависит лишь от поведения процесса до момента времени <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>. </p><p>Пусть теперь </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 =\sup\{n\geq 0\mid Y_{n}\geq L\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo>≥</mo> <mn>0</mn> <mo>∣</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≥</mo> <mi>L</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =\sup\{n\geq 0\mid Y_{n}\geq L\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd3cc743a5def91ef7f06b25e2adf1556108a7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.098ex; height:2.843ex;" alt="{\displaystyle \sigma =\sup\{n\geq 0\mid Y_{n}\geq L\}}"></span></dd></dl> <p>— момент последнего достижения процессом <span class="mwe-math-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 \{Y_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>Y</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 \{Y_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58a38bdd3c7cbd3a4b6362ec1a5e69a7d5766872" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.894ex; height:2.843ex;" alt="{\displaystyle \{Y_{n}\}}"></span> уровня <span class="mwe-math-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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span>. Тогда <span class="mwe-math-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> не является марковским моментом, ибо событие <span class="mwe-math-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 \leq n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>σ</mi> <mo>≤</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\sigma \leq n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e496b20a2af99671d2077eca92c1cd840d5ad6af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.148ex; height:2.843ex;" alt="{\displaystyle \{\sigma \leq n\}}"></span> предполагает знание поведения процесса в будущем. </p> <div class="mw-heading mw-heading2"><h2 id="Общий_случай"><span id=".D0.9E.D0.B1.D1.89.D0.B8.D0.B9_.D1.81.D0.BB.D1.83.D1.87.D0.B0.D0.B9"></span>Общий случай</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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&veaction=edit&section=3" title="Редактировать раздел «Общий случай»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%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&action=edit&section=3" title="Редактировать код раздела «Общий случай»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Пусть дано <a href="/wiki/%D0%92%D0%B5%D1%80%D0%BE%D1%8F%D1%82%D0%BD%D0%BE%D1%81%D1%82%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Вероятностное пространство">вероятностное пространство</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 (\Omega ,{\mathcal {F}},\mathbb {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">P</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb8743f7565082ed1a9ee0490d9d71be82eafaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.902ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}"></span> с фильтрацией <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\mathcal {F}}_{t}\}_{t\in T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" 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 fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{\mathcal {F}}_{t}\}_{t\in T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/502d06196a101b3d5fc05c9d6af22e88e8138a32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.901ex; height:2.843ex;" alt="{\displaystyle \{{\mathcal {F}}_{t}\}_{t\in T}}"></span>, где <span class="mwe-math-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\subset [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\subset [0,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f29f3140d31c8efe0c403948c6af20d96cb97bd9" 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\subset [0,\infty )}"></span>. Тогда случайная величина <span class="mwe-math-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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> принимающая значения в <span class="mwe-math-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\cup \{\infty \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <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 T\cup \{\infty \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/797f5db5b04b242bf23c046f93b2cea3075cac63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.868ex; height:2.843ex;" alt="{\displaystyle T\cup \{\infty \}}"></span> называется марковским моментом относительно данной фильтрации, если <span class="mwe-math-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 \leq t\}\in {\mathcal {F}}_{t},\quad \forall t\in 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> <mo>,</mo> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\tau \leq t\}\in {\mathcal {F}}_{t},\quad \forall t\in T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dff6fde22549e84414cd93aba73703b5246025a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.768ex; height:2.843ex;" alt="{\displaystyle \{\tau \leq t\}\in {\mathcal {F}}_{t},\quad \forall t\in T}"></span>.</li></ul> <ul><li>Если дан процесс <span class="mwe-math-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_{t}\}_{t\in T}}"> <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"> <mi>t</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X_{t}\}_{t\in T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18d29bc7ff1908644cc092f09185407eb6391d86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.154ex; height:2.843ex;" alt="{\displaystyle \{X_{t}\}_{t\in T}}"></span>, и <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{t}=\sigma (X_{s}\mid s\leq t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mo>=</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>∣</mo> <mi>s</mi> <mo>≤</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{t}=\sigma (X_{s}\mid s\leq t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc1f330f8227aab2de7f294c5f9f0432ad45e77d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.628ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}_{t}=\sigma (X_{s}\mid s\leq t)}"></span> — его естественные σ-алгебры, то говорят, что <span class="mwe-math-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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> — марковский момент относительно процесса <span class="mwe-math-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_{t}\}}"> <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"> <mi>t</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X_{t}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aa498a302423f64556e0783cd198ee7def541bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.075ex; height:2.843ex;" alt="{\displaystyle \{X_{t}\}}"></span>.</li></ul> <ul><li>Марковский момент называется моментом остановки, если он конечен почти наверное, то есть</li></ul> <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 \mathbb {P} (\tau <\infty )=1}"> <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>τ</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (\tau <\infty )=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff33491598d2ff94be67c27658fdd736cf075ce8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.115ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (\tau <\infty )=1}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Свойства"><span id=".D0.A1.D0.B2.D0.BE.D0.B9.D1.81.D1.82.D0.B2.D0.B0"></span>Свойства</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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&veaction=edit&section=4" title="Редактировать раздел «Свойства»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%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&action=edit&section=4" title="Редактировать код раздела «Свойства»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Если <span class="mwe-math-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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> и <span class="mwe-math-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> — марковские моменты, то </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau +\sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>+</mo> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau +\sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c19128dadb06f65ab6e6a5af63ec1034f2b0bef0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.372ex; height:2.176ex;" alt="{\displaystyle \tau +\sigma }"></span> — марковский момент;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau \wedge \sigma \equiv \min(\tau ,\sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>∧</mo> <mi>σ</mi> <mo>≡</mo> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau \wedge \sigma \equiv \min(\tau ,\sigma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/888302af114efff37a11bb8f0a3a0b29bd830390" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.463ex; height:2.843ex;" alt="{\displaystyle \tau \wedge \sigma \equiv \min(\tau ,\sigma )}"></span> — марковский момент;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau \vee \sigma \equiv \max(\tau ,\sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>∨</mo> <mi>σ</mi> <mo>≡</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau \vee \sigma \equiv \max(\tau ,\sigma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4786f6828129ca9566768e6b3bfd8aca0555039" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.913ex; height:2.843ex;" alt="{\displaystyle \tau \vee \sigma \equiv \max(\tau ,\sigma )}"></span> — марковский момент.</li></ul> <p><i>Замечание</i>: момент остановки может не иметь конечного математического ожидания. </p> <div class="mw-heading mw-heading3"><h3 id="Пример_2"><span id=".D0.9F.D1.80.D0.B8.D0.BC.D0.B5.D1.80_2"></span>Пример</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%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&veaction=edit&section=5" title="Редактировать раздел «Пример»" class="mw-editsection-visualeditor"><span>править</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%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&action=edit&section=5" title="Редактировать код раздела «Пример»"><span>править код</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Пусть <span class="mwe-math-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_{t}\}_{t\geq 0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{W_{t}\}_{t\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc552386cec25bffe6ff93d795eb0663c5a09a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.271ex; height:2.843ex;" alt="{\displaystyle \{W_{t}\}_{t\geq 0}}"></span> — стандартный <a href="/wiki/%D0%92%D0%B8%D0%BD%D0%B5%D1%80%D0%BE%D0%B2%D1%81%D0%BA%D0%B8%D0%B9_%D0%BF%D1%80%D0%BE%D1%86%D0%B5%D1%81%D1%81" title="Винеровский процесс">винеровский процесс</a>. Пусть <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edd4f784b6e8bb68fa774213ceacbab2d97825dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >0}"></span>. Определим </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 \tau =\inf\{t\geq 0\mid W_{t}\geq \alpha \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">inf</mo> <mo fence="false" stretchy="false">{</mo> <mi>t</mi> <mo>≥</mo> <mn>0</mn> <mo>∣</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>≥</mo> <mi>α</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =\inf\{t\geq 0\mid W_{t}\geq \alpha \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a492435a3ce2ba9f5685e1361edf07f702db6885" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.92ex; height:2.843ex;" alt="{\displaystyle \tau =\inf\{t\geq 0\mid W_{t}\geq \alpha \}}"></span>.</dd></dl> <p>Тогда <span class="mwe-math-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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> — марковский момент, имеющий распределение, задаваемое плотностью вероятности </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_{\tau }(t)={\frac {\alpha }{\sqrt {2\pi t^{3}}}}e^{-{\frac {\alpha ^{2}}{2t}}},\quad t\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <msqrt> <mn>2</mn> <mi>π</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\tau }(t)={\frac {\alpha }{\sqrt {2\pi t^{3}}}}e^{-{\frac {\alpha ^{2}}{2t}}},\quad t\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e930b2cb899daf227f55bb55c4caa7f686c8ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:28.671ex; height:6.509ex;" alt="{\displaystyle f_{\tau }(t)={\frac {\alpha }{\sqrt {2\pi t^{3}}}}e^{-{\frac {\alpha ^{2}}{2t}}},\quad t\geq 0}"></span>.</dd></dl> <p>В частности <span class="mwe-math-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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> — момент остановки. Однако, </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 \mathbb {E} \tau =\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mi>τ</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} \tau =\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19fa317dc65ca095456278a3494dda2700b1b40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.174ex; height:2.176ex;" alt="{\displaystyle \mathbb {E} \tau =\infty }"></span>.</dd></dl> <div role="navigation" class="navbox" aria-labelledby="Ссылки_на_внешние_ресурсы" 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style="width:52px"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Question_book-4.svg/48px-Question_book-4.svg.png" decoding="async" width="48" height="37" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Question_book-4.svg/72px-Question_book-4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Question_book-4.svg/96px-Question_book-4.svg.png 2x" data-file-width="262" data-file-height="204" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-div">В статье <b>не хватает <a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D1%81%D1%8B%D0%BB%D0%BA%D0%B8_%D0%BD%D0%B0_%D0%B8%D1%81%D1%82%D0%BE%D1%87%D0%BD%D0%B8%D0%BA%D0%B8" title="Википедия:Ссылки на источники">ссылок на источники</a></b> (см. <a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%9F%D0%BE%D0%B8%D1%81%D0%BA_%D0%B8%D1%81%D1%82%D0%BE%D1%87%D0%BD%D0%B8%D0%BA%D0%BE%D0%B2" title="Википедия:Поиск источников">рекомендации по поиску</a>).</div><div class="mbox-textsmall-div hide-when-compact"><span class="hide-when-compact"> Информация должна быть <a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%9F%D1%80%D0%BE%D0%B2%D0%B5%D1%80%D1%8F%D0%B5%D0%BC%D0%BE%D1%81%D1%82%D1%8C" title="Википедия:Проверяемость">проверяема</a>, иначе она может быть удалена. Вы можете <a class="external text" href="https://ru.wikipedia.org/w/index.php?title=%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&action=edit">отредактировать</a> статью, добавив ссылки на <a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%90%D0%B2%D1%82%D0%BE%D1%80%D0%B8%D1%82%D0%B5%D1%82%D0%BD%D1%8B%D0%B5_%D0%B8%D1%81%D1%82%D0%BE%D1%87%D0%BD%D0%B8%D0%BA%D0%B8" title="Википедия:Авторитетные источники">авторитетные источники</a> в виде <a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D0%BD%D0%BE%D1%81%D0%BA%D0%B8" title="Википедия:Сноски">сносок</a>.</span> <span class="mbox-date"><i>(<span class="date">8 июня 2019</span>)</i></span></div></td></tr></tbody></table></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="">Источник — <a dir="ltr" href="https://ru.wikipedia.org/w/index.php?title=Марковский_момент_времени&oldid=140307552">https://ru.wikipedia.org/w/index.php?title=Марковский_момент_времени&oldid=140307552</a></div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D0%B8" title="Служебная:Категории">Категория</a>: <ul><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D1%8B%D0%B5_%D0%BF%D1%80%D0%BE%D1%86%D0%B5%D1%81%D1%81%D1%8B" title="Категория:Случайные процессы">Случайные процессы</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Скрытые категории: <ul><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D1%82%D0%B0%D1%82%D1%8C%D0%B8_%D0%B1%D0%B5%D0%B7_%D1%81%D1%81%D1%8B%D0%BB%D0%BE%D0%BA_%D0%BD%D0%B0_%D0%B8%D1%81%D1%82%D0%BE%D1%87%D0%BD%D0%B8%D0%BA%D0%B8_%D1%81_%D0%B8%D1%8E%D0%BD%D1%8F_2019_%D0%B3%D0%BE%D0%B4%D0%B0" title="Категория:Википедия:Статьи без ссылок на источники с июня 2019 года">Википедия:Статьи без ссылок на источники с июня 2019 года</a></li><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D1%82%D0%B0%D1%82%D1%8C%D0%B8_%D0%B1%D0%B5%D0%B7_%D0%B8%D1%81%D1%82%D0%BE%D1%87%D0%BD%D0%B8%D0%BA%D0%BE%D0%B2_(%D1%82%D0%B8%D0%BF:_%D0%BD%D0%B5_%D1%83%D0%BA%D0%B0%D0%B7%D0%B0%D0%BD)" title="Категория:Википедия:Статьи без источников (тип: не указан)">Википедия:Статьи без источников (тип: не указан)</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Навигация</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Персональные инструменты</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="Страница участника для моего IP">Вы не представились системе</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%9C%D0%BE%D1%91_%D0%BE%D0%B1%D1%81%D1%83%D0%B6%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5" title="Страница обсуждений для моего IP [n]" accesskey="n"><span>Обсуждение</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%9C%D0%BE%D0%B9_%D0%B2%D0%BA%D0%BB%D0%B0%D0%B4" title="Список правок, сделанных с этого IP-адреса [y]" accesskey="y"><span>Вклад</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%A1%D0%BE%D0%B7%D0%B4%D0%B0%D1%82%D1%8C_%D1%83%D1%87%D1%91%D1%82%D0%BD%D1%83%D1%8E_%D0%B7%D0%B0%D0%BF%D0%B8%D1%81%D1%8C&returnto=%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="Мы предлагаем вам создать учётную запись и войти в систему, хотя это и не обязательно."><span>Создать учётную запись</span></a></li><li id="pt-login" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%92%D1%85%D0%BE%D0%B4&returnto=%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="Здесь можно зарегистрироваться в системе, но это необязательно. [o]" accesskey="o"><span>Войти</span></a></li> </ul> </div> </nav> <div id="left-navigation"> <nav id="p-namespaces" class="mw-portlet mw-portlet-namespaces vector-menu-tabs vector-menu-tabs-legacy vector-menu" aria-labelledby="p-namespaces-label" > <h3 id="p-namespaces-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Пространства имён</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected mw-list-item"><a href="/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="Просмотреть контентную страницу [c]" accesskey="c"><span>Статья</span></a></li><li id="ca-talk" class="new mw-list-item"><a href="/w/index.php?title=%D0%9E%D0%B1%D1%81%D1%83%D0%B6%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5:%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&action=edit&redlink=1" rel="discussion" class="new" title="Обсуждение основной страницы (страница отсутствует) [t]" accesskey="t"><span>Обсуждение</span></a></li> </ul> </div> </nav> <nav id="p-variants" class="mw-portlet mw-portlet-variants emptyPortlet vector-menu-dropdown vector-menu" aria-labelledby="p-variants-label" > <input type="checkbox" id="p-variants-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-variants" class="vector-menu-checkbox" aria-labelledby="p-variants-label" > <label id="p-variants-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">русский</span> </label> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </nav> </div> <div id="right-navigation"> <nav id="p-views" class="mw-portlet mw-portlet-views vector-menu-tabs vector-menu-tabs-legacy vector-menu" aria-labelledby="p-views-label" > <h3 id="p-views-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Просмотры</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected mw-list-item"><a href="/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"><span>Читать</span></a></li><li id="ca-ve-edit" class="mw-list-item"><a href="/w/index.php?title=%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&veaction=edit" title="Редактировать данную страницу [v]" accesskey="v"><span>Править</span></a></li><li id="ca-edit" class="collapsible mw-list-item"><a href="/w/index.php?title=%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&action=edit" title="Править исходный текст этой страницы [e]" accesskey="e"><span>Править код</span></a></li><li id="ca-history" class="mw-list-item"><a href="/w/index.php?title=%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&action=history" title="Журнал изменений страницы [h]" accesskey="h"><span>История</span></a></li> </ul> </div> </nav> <nav id="p-cactions" class="mw-portlet mw-portlet-cactions emptyPortlet vector-menu-dropdown vector-menu" aria-labelledby="p-cactions-label" title="Больше возможностей" > <input type="checkbox" id="p-cactions-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-cactions" class="vector-menu-checkbox" aria-labelledby="p-cactions-label" > <label id="p-cactions-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Ещё</span> </label> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </nav> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <h3 >Поиск</h3> <form action="/w/index.php" id="searchform" class="vector-search-box-form"> <div id="simpleSearch" class="vector-search-box-inner" data-search-loc="header-navigation"> <input class="vector-search-box-input" type="search" name="search" placeholder="Искать в Википедии" aria-label="Искать в Википедии" autocapitalize="sentences" title="Искать в Википедии [f]" accesskey="f" id="searchInput" > <input type="hidden" name="title" value="Служебная:Поиск"> <input id="mw-searchButton" class="searchButton mw-fallbackSearchButton" type="submit" name="fulltext" title="Найти страницы, содержащие указанный текст" value="Найти"> <input id="searchButton" class="searchButton" type="submit" name="go" title="Перейти к странице, имеющей в точности такое название" value="Перейти"> </div> </form> </div> </div> </div> <div id="mw-panel" class="vector-legacy-sidebar"> <div id="p-logo" role="banner"> <a class="mw-wiki-logo" href="/wiki/%D0%97%D0%B0%D0%B3%D0%BB%D0%B0%D0%B2%D0%BD%D0%B0%D1%8F_%D1%81%D1%82%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D0%B0" title="Перейти на заглавную страницу"></a> </div> <nav id="p-navigation" class="mw-portlet mw-portlet-navigation vector-menu-portal portal vector-menu" aria-labelledby="p-navigation-label" > <h3 id="p-navigation-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Навигация</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/%D0%97%D0%B0%D0%B3%D0%BB%D0%B0%D0%B2%D0%BD%D0%B0%D1%8F_%D1%81%D1%82%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D0%B0" title="Перейти на заглавную страницу [z]" accesskey="z"><span>Заглавная страница</span></a></li><li id="n-content" class="mw-list-item"><a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D0%BE%D0%B4%D0%B5%D1%80%D0%B6%D0%B0%D0%BD%D0%B8%D0%B5"><span>Содержание</span></a></li><li id="n-featured" class="mw-list-item"><a href="/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%98%D0%B7%D0%B1%D1%80%D0%B0%D0%BD%D0%BD%D1%8B%D0%B5_%D1%81%D1%82%D0%B0%D1%82%D1%8C%D0%B8" title="Статьи, считающиеся лучшими статьями проекта"><span>Избранные статьи</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D0%B0%D1%8F_%D1%81%D1%82%D1%80%D0%B0%D0%BD%D0%B8%D1%86%D0%B0" title="Посмотреть случайно выбранную страницу [x]" accesskey="x"><span>Случайная статья</span></a></li><li id="n-currentevents" class="mw-list-item"><a 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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 — английский" lang="en" hreflang="en" data-title="Stopping time" data-language-autonym="English" data-language-local-name="английский" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Tiempo_de_parada" title="Tiempo de parada — испанский" lang="es" hreflang="es" data-title="Tiempo de parada" data-language-autonym="Español" data-language-local-name="испанский" 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%B2%D9%85%D8%A7%D9%86_%D8%AA%D9%88%D9%82%D9%81" title="زمان توقف — персидский" lang="fa" hreflang="fa" data-title="زمان توقف" data-language-autonym="فارسی" data-language-local-name="персидский" 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 — французский" lang="fr" hreflang="fr" data-title="Temps d'arrêt" data-language-autonym="Français" data-language-local-name="французский" 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 — иврит" lang="he" hreflang="he" data-title="זמן עצירה" data-language-autonym="עברית" data-language-local-name="иврит" 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 — итальянский" lang="it" hreflang="it" data-title="Tempo di arresto" data-language-autonym="Italiano" data-language-local-name="итальянский" 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 — корейский" lang="ko" hreflang="ko" data-title="정지 시간" data-language-autonym="한국어" data-language-local-name="корейский" 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 — нидерландский" lang="nl" hreflang="nl" data-title="Stoptijd" data-language-autonym="Nederlands" data-language-local-name="нидерландский" 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 — польский" lang="pl" hreflang="pl" data-title="Moment zatrzymania" data-language-autonym="Polski" data-language-local-name="польский" class="interlanguage-link-target"><span>Polski</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 — украинский" lang="uk" hreflang="uk" data-title="Марковський момент часу" data-language-autonym="Українська" data-language-local-name="украинский" 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 — китайский" lang="zh" hreflang="zh" data-title="停时" data-language-autonym="中文" data-language-local-name="китайский" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit 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