Renewal theory - Wikipedia

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id="toc-Renewal_processes-sublist" class="vector-toc-list"> <li id="toc-Introduction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Introduction"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Introduction</span> </div> </a> <ul id="toc-Introduction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formal_definition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formal_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Formal definition</span> </div> </a> <ul id="toc-Formal_definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interpretation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interpretation"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Interpretation</span> </div> </a> <ul id="toc-Interpretation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Renewal-reward_processes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Renewal-reward_processes"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Renewal-reward processes</span> </div> </a> <button aria-controls="toc-Renewal-reward_processes-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Renewal-reward processes subsection</span> </button> <ul id="toc-Renewal-reward_processes-sublist" class="vector-toc-list"> <li id="toc-Interpretation_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interpretation_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Interpretation</span> </div> </a> <ul id="toc-Interpretation_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Renewal_function" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Renewal_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Renewal function</span> </div> </a> <button aria-controls="toc-Renewal_function-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Renewal function subsection</span> </button> <ul id="toc-Renewal_function-sublist" class="vector-toc-list"> <li id="toc-Elementary_renewal_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elementary_renewal_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Elementary renewal theorem</span> </div> </a> <ul id="toc-Elementary_renewal_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elementary_renewal_theorem_for_renewal_reward_processes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elementary_renewal_theorem_for_renewal_reward_processes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Elementary renewal theorem for renewal reward processes</span> </div> </a> <ul id="toc-Elementary_renewal_theorem_for_renewal_reward_processes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Renewal_equation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Renewal_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Renewal equation</span> </div> </a> <ul id="toc-Renewal_equation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Key_renewal_theorem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Key_renewal_theorem"> <div 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class="vector-toc-numb">5</span> <span>Asymptotic properties</span> </div> </a> <ul id="toc-Asymptotic_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inspection_paradox" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Inspection_paradox"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Inspection paradox</span> </div> </a> <ul id="toc-Inspection_paradox-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Superposition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Superposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Superposition</span> </div> </a> <ul id="toc-Superposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_application" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" 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Instead of <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponentially distributed</a> holding times, a renewal process may have any <a href="/wiki/IID" class="mw-redirect" title="IID">independent and identically distributed</a> (IID) holding times that have finite mean. A renewal-reward process additionally has a random sequence of rewards incurred at each holding time, which are IID but need not be independent of the holding times. </p><p>A renewal process has asymptotic properties analogous to the <a href="/wiki/Strong_law_of_large_numbers" class="mw-redirect" title="Strong law of large numbers">strong law of large numbers</a> and <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a>. The renewal function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea26578dd72bf4dbc3fa391c9feb11eed495699b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.689ex; height:2.843ex;" alt="{\displaystyle m(t)}"></span> (expected number of arrivals) and reward function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b84f700860ee7af27797d11ddfad3d185eb7af0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.765ex; height:2.843ex;" alt="{\displaystyle g(t)}"></span> (expected reward value) are of key importance in renewal theory. The renewal function satisfies a recursive integral equation, the renewal equation. The key renewal equation gives the limiting value of the <a href="/wiki/Convolution" title="Convolution">convolution</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m'(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>m</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m'(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3760df447bcdc752feb0726ad47d48c6f1203be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.374ex; height:3.009ex;" alt="{\displaystyle m'(t)}"></span> with a suitable non-negative function. The superposition of renewal processes can be studied as a special case of <a href="/wiki/Markov_renewal_process" title="Markov renewal process">Markov renewal processes</a>. </p><p>Applications include calculating the best strategy for replacing worn-out machinery in a factory and comparing the long-term benefits of different insurance policies. The inspection paradox relates to the fact that observing a renewal interval at time <i>t</i> gives an interval with average value larger than that of an average renewal interval. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Renewal_processes">Renewal processes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Renewal_theory&action=edit&section=1" title="Edit section: Renewal processes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Introduction">Introduction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Renewal_theory&action=edit&section=2" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>renewal process</b> is a generalization of the <a href="/wiki/Poisson_process" class="mw-redirect" title="Poisson process">Poisson process</a>. In essence, the Poisson process is a <a href="/wiki/Continuous-time_Markov_process" class="mw-redirect" title="Continuous-time Markov process">continuous-time Markov process</a> on the positive integers (usually starting at zero) which has independent <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponentially distributed</a> holding times at each integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> before advancing to the next integer, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fe1bfc8314922e4c3fdb4e8eceb20a00b4f011d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.805ex; height:2.343ex;" alt="{\displaystyle i+1}"></span>. In a renewal process, the holding times need not have an exponential distribution; rather, the holding times may have any distribution on the positive numbers, so long as the holding times are independent and identically distributed (<a href="/wiki/IID" class="mw-redirect" title="IID">IID</a>) and have finite mean. </p> <div class="mw-heading mw-heading3"><h3 id="Formal_definition">Formal definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Renewal_theory&action=edit&section=3" title="Edit section: Formal definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Renewal_process.reetep.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Renewal_process.reetep.png/350px-Renewal_process.reetep.png" decoding="async" width="350" height="206" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Renewal_process.reetep.png/525px-Renewal_process.reetep.png 1.5x, //upload.wikimedia.org/wikipedia/commons/5/55/Renewal_process.reetep.png 2x" data-file-width="562" data-file-height="331" /></a><figcaption>Sample evolution of a renewal process with <i>holding times</i> <i>S</i><sub><i>i</i></sub> and jump times <i>J</i><sub><i>n</i></sub>.</figcaption></figure> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S_{i})_{i\geq 1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>≥</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S_{i})_{i\geq 1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c31a1bc09282fc43bd980f21e253f7981e1c0b81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.934ex; height:2.843ex;" alt="{\displaystyle (S_{i})_{i\geq 1}}"></span> be a sequence of positive <a href="/wiki/Independent_identically_distributed" class="mw-redirect" title="Independent identically distributed">independent identically distributed</a> <a href="/wiki/Random_variable" title="Random variable">random variables</a> with finite <a href="/wiki/Expected_value" title="Expected value">expected value</a> </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 0<\operatorname {E} [S_{i}]<\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<\operatorname {E} [S_{i}]<\infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/303b232700cf356843c58010dcd3393627c46153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.431ex; height:2.843ex;" alt="{\displaystyle 0<\operatorname {E} [S_{i}]<\infty .}"></span></dd></dl> <p>We refer to the random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de6e810a93f67802ecb603ee0e3324005c6e583e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.509ex;" alt="{\displaystyle S_{i}}"></span> as the "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-th holding time". </p><p>Define for each <i>n</i> > 0 : </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 J_{n}=\sum _{i=1}^{n}S_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{n}=\sum _{i=1}^{n}S_{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14c12ce60f166d69b8c0db3e1280de87c427863f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.221ex; height:6.843ex;" alt="{\displaystyle J_{n}=\sum _{i=1}^{n}S_{i},}"></span></dd></dl> <p>each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22b79b47e08e4e16510d309639ef56a24c28696c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.509ex; height:2.509ex;" alt="{\displaystyle J_{n}}"></span> is referred to as the "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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>-th jump time" and the intervals <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [J_{n},J_{n+1}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [J_{n},J_{n+1}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/817b42a083e1bc24d8215e47b2326c3a50cd86a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.445ex; height:2.843ex;" alt="{\displaystyle [J_{n},J_{n+1}]}"></span> are called "renewal intervals". </p><p>Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{t})_{t\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>t</mi> </mrow> </msub> <msub> <mo 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 (X_{t})_{t\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21296940fbeeae13b27893bdb77c0ae6a02ad23f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.486ex; height:2.843ex;" alt="{\displaystyle (X_{t})_{t\geq 0}}"></span> is given by random variable </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 X_{t}=\sum _{n=1}^{\infty }\operatorname {\mathbb {I} } _{\{J_{n}\leq t\}}=\sup \left\{\,n:J_{n}\leq t\,\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mrow> <mo>{</mo> <mrow> <mspace width="thinmathspace" /> <mi>n</mi> <mo>:</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <mi>t</mi> <mspace width="thinmathspace" /> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{t}=\sum _{n=1}^{\infty }\operatorname {\mathbb {I} } _{\{J_{n}\leq t\}}=\sup \left\{\,n:J_{n}\leq t\,\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/810be93c18275be0c5fe46ab5fef9df4563d4e30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.986ex; height:6.843ex;" alt="{\displaystyle X_{t}=\sum _{n=1}^{\infty }\operatorname {\mathbb {I} } _{\{J_{n}\leq t\}}=\sup \left\{\,n:J_{n}\leq t\,\right\}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {\mathbb {I} } _{\{J_{n}\leq t\}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {\mathbb {I} } _{\{J_{n}\leq t\}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa377357297a05d5e44eba5ec99b4a7045ee30b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.53ex; height:3.009ex;" alt="{\displaystyle \operatorname {\mathbb {I} } _{\{J_{n}\leq t\}}}"></span> is the <a href="/wiki/Indicator_function" title="Indicator function">indicator function</a> </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 \operatorname {\mathbb {I} } _{\{J_{n}\leq t\}}={\begin{cases}1,&{\text{if }}J_{n}\leq t\\0,&{\text{otherwise}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {\mathbb {I} } _{\{J_{n}\leq t\}}={\begin{cases}1,&{\text{if }}J_{n}\leq t\\0,&{\text{otherwise}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/027d2d6859c9fad3582fc012856773a5eea6f03d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.832ex; height:6.176ex;" alt="{\displaystyle \operatorname {\mathbb {I} } _{\{J_{n}\leq t\}}={\begin{cases}1,&{\text{if }}J_{n}\leq t\\0,&{\text{otherwise}}\end{cases}}}"></span></dd></dl> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{t})_{t\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>t</mi> </mrow> </msub> <msub> <mo 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 (X_{t})_{t\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21296940fbeeae13b27893bdb77c0ae6a02ad23f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.486ex; height:2.843ex;" alt="{\displaystyle (X_{t})_{t\geq 0}}"></span> represents the number of jumps that have occurred by time <i>t</i>, and is called a renewal process. </p> <div class="mw-heading mw-heading3"><h3 id="Interpretation">Interpretation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Renewal_theory&action=edit&section=4" title="Edit section: Interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If one considers events occurring at random times, one may choose to think of the holding times <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{S_{i}:i\geq 1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:</mo> <mi>i</mi> <mo>≥</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{S_{i}:i\geq 1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/041d730f4e38203b9c19e53c7f36460d23a94d94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.55ex; height:2.843ex;" alt="{\displaystyle \{S_{i}:i\geq 1\}}"></span> as the random time elapsed between two consecutive events. For example, if the renewal process is modelling the numbers of breakdown of different machines, then the holding time represents the time between one machine breaking down before another one does. </p><p>The Poisson process is the unique renewal process with the <a href="/wiki/Markov_property" title="Markov property">Markov property</a>,<sup id="cite_ref-FOOTNOTEGrimmettStirzaker1992393_1-0" class="reference"><a href="#cite_note-FOOTNOTEGrimmettStirzaker1992393-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> as the exponential distribution is the unique continuous random variable with the property of memorylessness. </p> <div class="mw-heading mw-heading2"><h2 id="Renewal-reward_processes">Renewal-reward processes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Renewal_theory&action=edit&section=5" title="Edit section: Renewal-reward processes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Renewal-reward_process.reetep.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Renewal-reward_process.reetep.png/350px-Renewal-reward_process.reetep.png" decoding="async" width="350" height="206" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Renewal-reward_process.reetep.png/525px-Renewal-reward_process.reetep.png 1.5x, //upload.wikimedia.org/wikipedia/commons/6/6e/Renewal-reward_process.reetep.png 2x" data-file-width="562" data-file-height="331" /></a><figcaption>Sample evolution of a renewal-reward process with <i>holding times</i> <i>S</i><sub><i>i</i></sub>, jump times <i>J</i><sub><i>n</i></sub> and rewards <i>W</i><sub><i>i</i></sub></figcaption></figure> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{1},W_{2},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{1},W_{2},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99c7ddf76125318166e87e8802170398948e5fea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.287ex; height:2.509ex;" alt="{\displaystyle W_{1},W_{2},\ldots }"></span> be a sequence of <a href="/wiki/IID" class="mw-redirect" title="IID">IID</a> random variables (<i>rewards</i>) satisfying </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 \operatorname {E} |W_{i}|<\infty .\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} |W_{i}|<\infty .\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50032855a131846466dd198242a5c220dea79ab8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.713ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} |W_{i}|<\infty .\,}"></span></dd></dl> <p>Then the random variable </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 Y_{t}=\sum _{i=1}^{X_{t}}W_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </munderover> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{t}=\sum _{i=1}^{X_{t}}W_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8479b191a66969ca916d3696c531725b9de4dfce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.01ex; height:7.343ex;" alt="{\displaystyle Y_{t}=\sum _{i=1}^{X_{t}}W_{i}}"></span></dd></dl> <p>is called a <b>renewal-reward process</b>. Note that unlike the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de6e810a93f67802ecb603ee0e3324005c6e583e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.509ex;" alt="{\displaystyle S_{i}}"></span>, each <span class="mwe-math-element"><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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7301a4cfd04d4f5db4549fdf23746a0d2ce9f387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle W_{i}}"></span> may take negative values as well as positive values. </p><p>The random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95734a78eb8407939c3496cbfd92763ced1e41e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.177ex; height:2.509ex;" alt="{\displaystyle Y_{t}}"></span> depends on two sequences: the holding times <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1},S_{2},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1},S_{2},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6de0f98822fbb68f9369708aaa1a711dd183175c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.749ex; height:2.509ex;" alt="{\displaystyle S_{1},S_{2},\ldots }"></span> and the rewards <span class="mwe-math-element"><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_{1},W_{2},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{1},W_{2},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99c7ddf76125318166e87e8802170398948e5fea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.287ex; height:2.509ex;" alt="{\displaystyle W_{1},W_{2},\ldots }"></span> These two sequences need not be independent. In particular, <span class="mwe-math-element"><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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7301a4cfd04d4f5db4549fdf23746a0d2ce9f387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle W_{i}}"></span> may be a function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de6e810a93f67802ecb603ee0e3324005c6e583e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.509ex;" alt="{\displaystyle S_{i}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Interpretation_2">Interpretation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Renewal_theory&action=edit&section=6" title="Edit section: Interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the context of the above interpretation of the holding times as the time between successive malfunctions of a machine, the "rewards" <span class="mwe-math-element"><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_{1},W_{2},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{1},W_{2},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99c7ddf76125318166e87e8802170398948e5fea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.287ex; height:2.509ex;" alt="{\displaystyle W_{1},W_{2},\ldots }"></span> (which in this case happen to be negative) may be viewed as the successive repair costs incurred as a result of the successive malfunctions. </p><p>An alternative analogy is that we have a magic goose which lays eggs at intervals (holding times) distributed as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de6e810a93f67802ecb603ee0e3324005c6e583e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.509ex;" alt="{\displaystyle S_{i}}"></span>. Sometimes it lays golden eggs of random weight, and sometimes it lays toxic eggs (also of random weight) which require responsible (and costly) disposal. The "rewards" <span class="mwe-math-element"><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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7301a4cfd04d4f5db4549fdf23746a0d2ce9f387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle W_{i}}"></span> are the successive (random) financial losses/gains resulting from successive eggs (<i>i</i> = 1,2,3,...) and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95734a78eb8407939c3496cbfd92763ced1e41e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.177ex; height:2.509ex;" alt="{\displaystyle Y_{t}}"></span> records the total financial "reward" at time <i>t</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Renewal_function">Renewal function</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Renewal_theory&action=edit&section=7" title="Edit section: Renewal function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We define the <b>renewal function</b> as the <a href="/wiki/Expected_value" title="Expected value">expected value</a> of the number of jumps observed up to some time <span class="mwe-math-element"><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/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>: </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 m(t)=\operatorname {E} [X_{t}].\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m(t)=\operatorname {E} [X_{t}].\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c55cdd0910db32f06a61d6638c5f1680bcfa8a46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.448ex; height:2.843ex;" alt="{\displaystyle m(t)=\operatorname {E} [X_{t}].\,}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Elementary_renewal_theorem">Elementary renewal theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Renewal_theory&action=edit&section=8" title="Edit section: Elementary renewal theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The renewal function satisfies </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{t\to \infty }{\frac {1}{t}}m(t)={\frac {1}{\operatorname {E} [S_{1}]}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>t</mi> </mfrac> </mrow> <mi>m</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{t\to \infty }{\frac {1}{t}}m(t)={\frac {1}{\operatorname {E} [S_{1}]}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f16c7b6c9599b43194e1122a8121150858b0e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.892ex; height:6.009ex;" alt="{\displaystyle \lim _{t\to \infty }{\frac {1}{t}}m(t)={\frac {1}{\operatorname {E} [S_{1}]}}.}"></span></dd></dl> <dl><dd><table class="toccolours collapsible collapsed" width="80%" style="text-align:left"> <tbody><tr> <th>Proof </th></tr> <tr> <td>The <a href="#Asymptotic_properties">strong law of large numbers for renewal processes</a> implies <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{t\to \infty }{\frac {X_{t}}{t}}={\frac {1}{\operatorname {E} [S_{1}]}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>t</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{t\to \infty }{\frac {X_{t}}{t}}={\frac {1}{\operatorname {E} [S_{1}]}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31e47c2a236e95b57ae74176253c46e734003077" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.791ex; height:6.009ex;" alt="{\displaystyle \lim _{t\to \infty }{\frac {X_{t}}{t}}={\frac {1}{\operatorname {E} [S_{1}]}}.}"></span></dd></dl> <p>To prove the elementary renewal theorem, it is sufficient to show that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\frac {X_{t}}{t}};t\geq 0\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>t</mi> </mfrac> </mrow> <mo>;</mo> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\frac {X_{t}}{t}};t\geq 0\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/310b9c6237053d1bc91a0f2245860ada9b74ffaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.207ex; height:6.176ex;" alt="{\displaystyle \left\{{\frac {X_{t}}{t}};t\geq 0\right\}}"></span> is uniformly integrable. </p><p>To do this, consider some truncated renewal process where the holding times are defined by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {S_{n}}}=a\operatorname {\mathbb {I} } \{S_{n}>a\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> </mrow> <mo>⁡</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>></mo> <mi>a</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {S_{n}}}=a\operatorname {\mathbb {I} } \{S_{n}>a\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de0a071d9f8ce6a4e56b4ce06076d92661aab6f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.675ex; height:3.509ex;" alt="{\displaystyle {\overline {S_{n}}}=a\operatorname {\mathbb {I} } \{S_{n}>a\}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> is a point such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<F(a)=p<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<F(a)=p<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06291b9dca041b9e62244b43465414a62f034461" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.57ex; height:2.843ex;" alt="{\displaystyle 0<F(a)=p<1}"></span> which exists for all non-deterministic renewal processes. This new renewal process <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {X}}_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {X}}_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61d5b440edc309ff486464feff8b26acb4cf1917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.957ex; height:3.343ex;" alt="{\displaystyle {\overline {X}}_{t}}"></span> is an upper bound on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </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/82120d04dfb3cbadc4912951dd12b5568c9cd8f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.75ex; height:2.509ex;" alt="{\displaystyle X_{t}}"></span> and its renewals can only occur on the lattice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{na;n\in \mathbb {N} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mi>a</mi> <mo>;</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{na;n\in \mathbb {N} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edad705e09c7b17c22b469dd8a70179dfdeaa534" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.897ex; height:2.843ex;" alt="{\displaystyle \{na;n\in \mathbb {N} \}}"></span>. Furthermore, the number of renewals at each time is geometric with parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>. So we have </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 {\begin{aligned}{\overline {X_{t}}}&\leq \sum _{i=1}^{[at]}\operatorname {Geometric} (p)\\\operatorname {E} \left[\,{\overline {X_{t}}}^{2}\,\right]&\leq C_{1}t+C_{2}t^{2}\\P\left({\frac {X_{t}}{t}}>x\right)&\leq {\frac {\operatorname {E} \left[X_{t}^{2}\right]}{t^{2}x^{2}}}\leq {\frac {\operatorname {E} \left[{\overline {X_{t}}}^{2}\right]}{t^{2}x^{2}}}\leq {\frac {C}{x^{2}}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>≤</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>a</mi> <mi>t</mi> <mo stretchy="false">]</mo> </mrow> </munderover> <mi>Geometric</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mrow> <mo>]</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>≤</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>t</mi> </mfrac> </mrow> <mo>></mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>]</mo> </mrow> </mrow> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>]</mo> </mrow> </mrow> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overline {X_{t}}}&\leq \sum _{i=1}^{[at]}\operatorname {Geometric} (p)\\\operatorname {E} \left[\,{\overline {X_{t}}}^{2}\,\right]&\leq C_{1}t+C_{2}t^{2}\\P\left({\frac {X_{t}}{t}}>x\right)&\leq {\frac {\operatorname {E} \left[X_{t}^{2}\right]}{t^{2}x^{2}}}\leq {\frac {\operatorname {E} \left[{\overline {X_{t}}}^{2}\right]}{t^{2}x^{2}}}\leq {\frac {C}{x^{2}}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c6c48d3d77404c6ada9e0bbc62a6219c76e96b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.005ex; width:43.425ex; height:21.176ex;" alt="{\displaystyle {\begin{aligned}{\overline {X_{t}}}&\leq \sum _{i=1}^{[at]}\operatorname {Geometric} (p)\\\operatorname {E} \left[\,{\overline {X_{t}}}^{2}\,\right]&\leq C_{1}t+C_{2}t^{2}\\P\left({\frac {X_{t}}{t}}>x\right)&\leq {\frac {\operatorname {E} \left[X_{t}^{2}\right]}{t^{2}x^{2}}}\leq {\frac {\operatorname {E} \left[{\overline {X_{t}}}^{2}\right]}{t^{2}x^{2}}}\leq {\frac {C}{x^{2}}}.\end{aligned}}}"></span></dd></dl> </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Elementary_renewal_theorem_for_renewal_reward_processes">Elementary renewal theorem for renewal reward processes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Renewal_theory&action=edit&section=9" title="Edit section: Elementary renewal theorem for renewal reward processes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We define the <b>reward function</b>: </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 g(t)=\operatorname {E} [Y_{t}].\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(t)=\operatorname {E} [Y_{t}].\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35d59e7c158d2dd5aab3baf64d637375dda9b1a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.95ex; height:2.843ex;" alt="{\displaystyle g(t)=\operatorname {E} [Y_{t}].\,}"></span></dd></dl> <p>The reward function satisfies </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{t\to \infty }{\frac {1}{t}}g(t)={\frac {\operatorname {E} [W_{1}]}{\operatorname {E} [S_{1}]}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>t</mi> </mfrac> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{t\to \infty }{\frac {1}{t}}g(t)={\frac {\operatorname {E} [W_{1}]}{\operatorname {E} [S_{1}]}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94fc2bbcc7f4205c065d3824a46b532a6a141ed1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.736ex; height:6.509ex;" alt="{\displaystyle \lim _{t\to \infty }{\frac {1}{t}}g(t)={\frac {\operatorname {E} [W_{1}]}{\operatorname {E} [S_{1}]}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Renewal_equation">Renewal equation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Renewal_theory&action=edit&section=10" title="Edit section: Renewal equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The renewal function satisfies </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 m(t)=F_{S}(t)+\int _{0}^{t}m(t-s)f_{S}(s)\,ds}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>m</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m(t)=F_{S}(t)+\int _{0}^{t}m(t-s)f_{S}(s)\,ds}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70304f10a3edfaf3d07193183c76c6f7823ee4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.386ex; height:6.176ex;" alt="{\displaystyle m(t)=F_{S}(t)+\int _{0}^{t}m(t-s)f_{S}(s)\,ds}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b869688c17fd16253b07d1d0d1abea06ddc8024c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.787ex; height:2.509ex;" alt="{\displaystyle F_{S}}"></span> is the cumulative distribution function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b44c97f83aebb50c3fd26b567ff9b005dc7b82b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.432ex; height:2.509ex;" alt="{\displaystyle f_{S}}"></span> is the corresponding probability density function. </p> <dl><dd><table class="toccolours collapsible collapsed" width="80%" style="text-align:left"> <tbody><tr> <th>Proof<sup id="cite_ref-FOOTNOTEGrimmettStirzaker1992390_2-0" class="reference"><a href="#cite_note-FOOTNOTEGrimmettStirzaker1992390-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </th></tr> <tr> <td>We may iterate the expectation about the first holding time: <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 m(t)=\operatorname {E} [X_{t}]=\operatorname {E} [\operatorname {E} (X_{t}\mid S_{1})].\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m(t)=\operatorname {E} [X_{t}]=\operatorname {E} [\operatorname {E} (X_{t}\mid S_{1})].\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e05d07a9e3171af23fba144fbcb5c2a03ce090b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.982ex; height:2.843ex;" alt="{\displaystyle m(t)=\operatorname {E} [X_{t}]=\operatorname {E} [\operatorname {E} (X_{t}\mid S_{1})].\,}"></span></dd></dl> <p>From the definition of the renewal process, we have </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 \operatorname {E} (X_{t}\mid S_{1}=s)=\operatorname {\mathbb {I} } _{\{t\geq s\}}\left(1+\operatorname {E} [X_{t-s}]\right).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mi>t</mi> <mo>≥</mo> <mi>s</mi> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−</mo> <mi>s</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (X_{t}\mid S_{1}=s)=\operatorname {\mathbb {I} } _{\{t\geq s\}}\left(1+\operatorname {E} [X_{t-s}]\right).\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c39464a2eaeb45121361d11d3aba4438b5a0a56a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:37.793ex; height:3.176ex;" alt="{\displaystyle \operatorname {E} (X_{t}\mid S_{1}=s)=\operatorname {\mathbb {I} } _{\{t\geq s\}}\left(1+\operatorname {E} [X_{t-s}]\right).\,}"></span></dd></dl> <p>So </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 {\begin{aligned}m(t)&=\operatorname {E} [X_{t}]\\[12pt]&=\operatorname {E} [\operatorname {E} (X_{t}\mid S_{1})]\\[12pt]&=\int _{0}^{\infty }\operatorname {E} (X_{t}\mid S_{1}=s)f_{S}(s)\,ds\\[12pt]&=\int _{0}^{\infty }\operatorname {\mathbb {I} } _{\{t\geq s\}}\left(1+\operatorname {E} [X_{t-s}]\right)f_{S}(s)\,ds\\[12pt]&=\int _{0}^{t}\left(1+m(t-s)\right)f_{S}(s)\,ds\\[12pt]&=F_{S}(t)+\int _{0}^{t}m(t-s)f_{S}(s)\,ds,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.5em 1.5em 1.5em 1.5em 1.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>m</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>s</mi> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mi>t</mi> <mo>≥</mo> <mi>s</mi> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−</mo> <mi>s</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>m</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>m</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}m(t)&=\operatorname {E} [X_{t}]\\[12pt]&=\operatorname {E} [\operatorname {E} (X_{t}\mid S_{1})]\\[12pt]&=\int _{0}^{\infty }\operatorname {E} (X_{t}\mid S_{1}=s)f_{S}(s)\,ds\\[12pt]&=\int _{0}^{\infty }\operatorname {\mathbb {I} } _{\{t\geq s\}}\left(1+\operatorname {E} [X_{t-s}]\right)f_{S}(s)\,ds\\[12pt]&=\int _{0}^{t}\left(1+m(t-s)\right)f_{S}(s)\,ds\\[12pt]&=F_{S}(t)+\int _{0}^{t}m(t-s)f_{S}(s)\,ds,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d1ec955a1a4580e78f07e747ff551c95c9e67f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -21.634ex; margin-bottom: -0.204ex; width:40.203ex; height:44.843ex;" alt="{\displaystyle {\begin{aligned}m(t)&=\operatorname {E} [X_{t}]\\[12pt]&=\operatorname {E} [\operatorname {E} (X_{t}\mid S_{1})]\\[12pt]&=\int _{0}^{\infty }\operatorname {E} (X_{t}\mid S_{1}=s)f_{S}(s)\,ds\\[12pt]&=\int _{0}^{\infty }\operatorname {\mathbb {I} } _{\{t\geq s\}}\left(1+\operatorname {E} [X_{t-s}]\right)f_{S}(s)\,ds\\[12pt]&=\int _{0}^{t}\left(1+m(t-s)\right)f_{S}(s)\,ds\\[12pt]&=F_{S}(t)+\int _{0}^{t}m(t-s)f_{S}(s)\,ds,\end{aligned}}}"></span></dd></dl> <p>as required. </p> </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Key_renewal_theorem">Key renewal theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Renewal_theory&action=edit&section=11" title="Edit section: Key renewal theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>X</i> be a renewal process with renewal function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea26578dd72bf4dbc3fa391c9feb11eed495699b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.689ex; height:2.843ex;" alt="{\displaystyle m(t)}"></span> and interrenewal mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span>. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:[0,\infty )\rightarrow [0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</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 g:[0,\infty )\rightarrow [0,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01ba3a4594c6c6410ba5a765d44f1b18b372e67d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.811ex; height:2.843ex;" alt="{\displaystyle g:[0,\infty )\rightarrow [0,\infty )}"></span> be a function satisfying: </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 \int _{0}^{\infty }g(t)\,dt<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\infty }g(t)\,dt<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/055e76bc9c6e3d76109b89342d07776a542183f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.356ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{\infty }g(t)\,dt<\infty }"></span></li> <li><i>g</i> is monotone and non-increasing</li></ul> <p>The key renewal theorem states that, as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b543f76f961ec3f52d78fa3d72c3d87a521dd3a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.777ex; height:2.009ex;" alt="{\displaystyle t\rightarrow \infty }"></span>:<sup id="cite_ref-FOOTNOTEGrimmettStirzaker1992395_3-0" class="reference"><a href="#cite_note-FOOTNOTEGrimmettStirzaker1992395-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </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 \int _{0}^{t}g(t-x)m'(x)\,dx\rightarrow {\frac {1}{\mu }}\int _{0}^{\infty }g(x)\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>m</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>μ</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{t}g(t-x)m'(x)\,dx\rightarrow {\frac {1}{\mu }}\int _{0}^{\infty }g(x)\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b497b69c7e3948f89394b85673fea9afbda314e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.562ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{t}g(t-x)m'(x)\,dx\rightarrow {\frac {1}{\mu }}\int _{0}^{\infty }g(x)\,dx}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Renewal_theorem">Renewal theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Renewal_theory&action=edit&section=12" title="Edit section: Renewal theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Considering <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=\mathbb {I} _{[0,h]}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>h</mi> <mo stretchy="false">]</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=\mathbb {I} _{[0,h]}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06f9950634ecf29371b59d08ac982251513a7929" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.77ex; height:3.176ex;" alt="{\displaystyle g(x)=\mathbb {I} _{[0,h]}(x)}"></span> for any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbddb7a5cca6170575e4e73e769fbb434c2a3d71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.6ex; height:2.176ex;" alt="{\displaystyle h>0}"></span> gives as a special case the renewal theorem:<sup id="cite_ref-FOOTNOTEFeller1971347–351_4-0" class="reference"><a href="#cite_note-FOOTNOTEFeller1971347–351-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </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 m(t+h)-m(t)\rightarrow {\frac {h}{\mu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>m</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mi>μ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m(t+h)-m(t)\rightarrow {\frac {h}{\mu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63b0373c2c4b777b38f56f6c000b96a64bf0e37e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.25ex; height:5.843ex;" alt="{\displaystyle m(t+h)-m(t)\rightarrow {\frac {h}{\mu }}}"></span> as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b543f76f961ec3f52d78fa3d72c3d87a521dd3a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.777ex; height:2.009ex;" alt="{\displaystyle t\rightarrow \infty }"></span></dd></dl> <p>The result can be proved using integral equations or by a <a href="/wiki/Coupling_(probability)" title="Coupling (probability)">coupling</a> argument.<sup id="cite_ref-FOOTNOTEGrimmettStirzaker1992394–5_5-0" class="reference"><a href="#cite_note-FOOTNOTEGrimmettStirzaker1992394–5-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Though a special case of the key renewal theorem, it can be used to deduce the full theorem, by considering step functions and then increasing sequences of step functions.<sup id="cite_ref-FOOTNOTEGrimmettStirzaker1992395_3-1" class="reference"><a href="#cite_note-FOOTNOTEGrimmettStirzaker1992395-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Asymptotic_properties">Asymptotic properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Renewal_theory&action=edit&section=13" title="Edit section: Asymptotic properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Renewal processes and renewal-reward processes have properties analogous to the <a href="/wiki/Strong_law_of_large_numbers" class="mw-redirect" title="Strong law of large numbers">strong law of large numbers</a>, which can be derived from the same theorem. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{t})_{t\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>t</mi> </mrow> </msub> <msub> <mo 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 (X_{t})_{t\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21296940fbeeae13b27893bdb77c0ae6a02ad23f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.486ex; height:2.843ex;" alt="{\displaystyle (X_{t})_{t\geq 0}}"></span> is a renewal process and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Y_{t})_{t\geq 0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <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> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Y_{t})_{t\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a52866b4fb8967514ea2b18a95a6a06348bc368" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.912ex; height:2.843ex;" alt="{\displaystyle (Y_{t})_{t\geq 0}}"></span> is a renewal-reward process then: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{t\to \infty }{\frac {1}{t}}X_{t}={\frac {1}{\operatorname {E} [S_{1}]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>t</mi> </mfrac> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{t\to \infty }{\frac {1}{t}}X_{t}={\frac {1}{\operatorname {E} [S_{1}]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/550b5cf1752eb6ccca0ac9d68570e41b53a22cb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.306ex; height:6.009ex;" alt="{\displaystyle \lim _{t\to \infty }{\frac {1}{t}}X_{t}={\frac {1}{\operatorname {E} [S_{1}]}}}"></span><sup id="cite_ref-FOOTNOTEGrimmettStirzaker1992394_6-0" class="reference"><a href="#cite_note-FOOTNOTEGrimmettStirzaker1992394-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{t\to \infty }{\frac {1}{t}}Y_{t}={\frac {1}{\operatorname {E} [S_{1}]}}\operatorname {E} [W_{1}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>t</mi> </mfrac> </mrow> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{t\to \infty }{\frac {1}{t}}Y_{t}={\frac {1}{\operatorname {E} [S_{1}]}}\operatorname {E} [W_{1}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be5a873d353b5b6397927d08a8de29a6c83fb528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.244ex; height:6.009ex;" alt="{\displaystyle \lim _{t\to \infty }{\frac {1}{t}}Y_{t}={\frac {1}{\operatorname {E} [S_{1}]}}\operatorname {E} [W_{1}]}"></span></dd></dl> <p>almost surely. </p> <dl><dd><table class="toccolours collapsible collapsed" width="80%" style="text-align:left"> <tbody><tr> <th>Proof </th></tr> <tr> <td>First consider <span class="mwe-math-element"><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\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>t</mi> </mrow> </msub> <msub> <mo 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 (X_{t})_{t\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21296940fbeeae13b27893bdb77c0ae6a02ad23f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.486ex; height:2.843ex;" alt="{\displaystyle (X_{t})_{t\geq 0}}"></span>. By definition we have: <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 J_{X_{t}}\leq t\leq J_{X_{t}+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </msub> <mo>≤</mo> <mi>t</mi> <mo>≤</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{X_{t}}\leq t\leq J_{X_{t}+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcdb96290a0139e0b8d404c789aa5095b8e21c60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.196ex; height:2.676ex;" alt="{\displaystyle J_{X_{t}}\leq t\leq J_{X_{t}+1}}"></span></dd></dl> <p>for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/248525429e9cd266f53ab8c52d17bc206c546060" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.101ex; height:2.343ex;" alt="{\displaystyle t\geq 0}"></span> and so </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 {\frac {J_{X_{t}}}{X_{t}}}\leq {\frac {t}{X_{t}}}\leq {\frac {J_{X_{t}+1}}{X_{t}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mfrac> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mfrac> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {J_{X_{t}}}{X_{t}}}\leq {\frac {t}{X_{t}}}\leq {\frac {J_{X_{t}+1}}{X_{t}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15a67a56f40c0ce610bee75cb5619f94f805d41a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.615ex; height:6.009ex;" alt="{\displaystyle {\frac {J_{X_{t}}}{X_{t}}}\leq {\frac {t}{X_{t}}}\leq {\frac {J_{X_{t}+1}}{X_{t}}}}"></span></dd></dl> <p>for all <i>t</i> ≥ 0. </p><p>Now since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<\operatorname {E} [S_{i}]<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<\operatorname {E} [S_{i}]<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91136b7100899241b29332a22ba9019a3fbf9d5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.784ex; height:2.843ex;" alt="{\displaystyle 0<\operatorname {E} [S_{i}]<\infty }"></span> we have: </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 X_{t}\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{t}\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730a7aad0c1642801cd0c087ab1e9824776e6d89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.688ex; height:2.509ex;" alt="{\displaystyle X_{t}\to \infty }"></span></dd></dl> <p>as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a34d7a61899d577d950881b4a44888d43f3fa93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.777ex; height:2.009ex;" alt="{\displaystyle t\to \infty }"></span> <a href="/wiki/Almost_surely" title="Almost surely">almost surely</a> (with probability 1). Hence: </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 {\frac {J_{X_{t}}}{X_{t}}}={\frac {J_{n}}{n}}={\frac {1}{n}}\sum _{i=1}^{n}S_{i}\to \operatorname {E} [S_{1}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {J_{X_{t}}}{X_{t}}}={\frac {J_{n}}{n}}={\frac {1}{n}}\sum _{i=1}^{n}S_{i}\to \operatorname {E} [S_{1}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96912bc2a8e290d5a14f2d904657d328c316a303" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.462ex; height:6.843ex;" alt="{\displaystyle {\frac {J_{X_{t}}}{X_{t}}}={\frac {J_{n}}{n}}={\frac {1}{n}}\sum _{i=1}^{n}S_{i}\to \operatorname {E} [S_{1}]}"></span></dd></dl> <p>almost surely (using the strong law of large numbers); similarly: </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 {\frac {J_{X_{t}+1}}{X_{t}}}={\frac {J_{X_{t}+1}}{X_{t}+1}}{\frac {X_{t}+1}{X_{t}}}={\frac {J_{n+1}}{n+1}}{\frac {n+1}{n}}\to \operatorname {E} [S_{1}]\cdot 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">→</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>⋅</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {J_{X_{t}+1}}{X_{t}}}={\frac {J_{X_{t}+1}}{X_{t}+1}}{\frac {X_{t}+1}{X_{t}}}={\frac {J_{n+1}}{n+1}}{\frac {n+1}{n}}\to \operatorname {E} [S_{1}]\cdot 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e51991076c7d41fa573be6ea2d19232d8363ec2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:52.12ex; height:6.009ex;" alt="{\displaystyle {\frac {J_{X_{t}+1}}{X_{t}}}={\frac {J_{X_{t}+1}}{X_{t}+1}}{\frac {X_{t}+1}{X_{t}}}={\frac {J_{n+1}}{n+1}}{\frac {n+1}{n}}\to \operatorname {E} [S_{1}]\cdot 1}"></span></dd></dl> <p>almost surely. </p><p>Thus (since <span class="mwe-math-element"><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/X_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t/X_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c325359c8a071aff12c6da6c7a270497bfbf5dd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.752ex; height:2.843ex;" alt="{\displaystyle t/X_{t}}"></span> is sandwiched between the two terms) </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 {\frac {1}{t}}X_{t}\to {\frac {1}{\operatorname {E} [S_{1}]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>t</mi> </mfrac> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{t}}X_{t}\to {\frac {1}{\operatorname {E} [S_{1}]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cac04b5d2c7be16d2921d39a8adb8198cdaf9074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.555ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{t}}X_{t}\to {\frac {1}{\operatorname {E} [S_{1}]}}}"></span></dd></dl> <p>almost surely.<sup id="cite_ref-FOOTNOTEGrimmettStirzaker1992395_3-2" class="reference"><a href="#cite_note-FOOTNOTEGrimmettStirzaker1992395-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Next consider <span class="mwe-math-element"><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_{t})_{t\geq 0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <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> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Y_{t})_{t\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a52866b4fb8967514ea2b18a95a6a06348bc368" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.912ex; height:2.843ex;" alt="{\displaystyle (Y_{t})_{t\geq 0}}"></span>. We have </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 {\frac {1}{t}}Y_{t}={\frac {X_{t}}{t}}{\frac {1}{X_{t}}}Y_{t}\to {\frac {1}{\operatorname {E} [S_{1}]}}\cdot \operatorname {E} [W_{1}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>t</mi> </mfrac> </mrow> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>t</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mfrac> </mrow> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{t}}Y_{t}={\frac {X_{t}}{t}}{\frac {1}{X_{t}}}Y_{t}\to {\frac {1}{\operatorname {E} [S_{1}]}}\cdot \operatorname {E} [W_{1}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0bda248d59a2abd99cd0df2a8b3c05bab05cfe5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.232ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{t}}Y_{t}={\frac {X_{t}}{t}}{\frac {1}{X_{t}}}Y_{t}\to {\frac {1}{\operatorname {E} [S_{1}]}}\cdot \operatorname {E} [W_{1}]}"></span></dd></dl> <p>almost surely (using the first result and using the law of large numbers on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95734a78eb8407939c3496cbfd92763ced1e41e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.177ex; height:2.509ex;" alt="{\displaystyle Y_{t}}"></span>). </p> </td></tr></tbody></table></dd></dl> <p>Renewal processes additionally have a property analogous to the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a>:<sup id="cite_ref-FOOTNOTEGrimmettStirzaker1992394_6-1" class="reference"><a href="#cite_note-FOOTNOTEGrimmettStirzaker1992394-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </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 {\frac {X_{t}-t/\mu }{\sqrt {t\sigma ^{2}/\mu ^{3}}}}\to {\mathcal {N}}(0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>−</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>μ</mi> </mrow> <msqrt> <mi>t</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {X_{t}-t/\mu }{\sqrt {t\sigma ^{2}/\mu ^{3}}}}\to {\mathcal {N}}(0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f52b4d927cbf8990bcd9c0ad1438f6ad48a7fc2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.06ex; height:7.009ex;" alt="{\displaystyle {\frac {X_{t}-t/\mu }{\sqrt {t\sigma ^{2}/\mu ^{3}}}}\to {\mathcal {N}}(0,1)}"></span></dd></dl> <p><span class="anchor" id="The_inspection_paradox"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Inspection_paradox">Inspection paradox</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Renewal_theory&action=edit&section=14" title="Edit section: Inspection paradox"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Inspection_paradox.reetep.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Inspection_paradox.reetep.png/350px-Inspection_paradox.reetep.png" decoding="async" width="350" height="206" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Inspection_paradox.reetep.png/525px-Inspection_paradox.reetep.png 1.5x, //upload.wikimedia.org/wikipedia/commons/0/0c/Inspection_paradox.reetep.png 2x" data-file-width="562" data-file-height="331" /></a><figcaption>The renewal interval determined by the random point <i>t</i> (shown in red) is stochastically larger than the first renewal interval.</figcaption></figure><style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/List_of_paradoxes#Mathematics" title="List of paradoxes">List of paradoxes § Mathematics</a></div> <p>A curious feature of renewal processes is that if we wait some predetermined time <i>t</i> and then observe how large the renewal interval containing <i>t</i> is, we should expect it to be typically larger than a renewal interval of average size. </p><p>Mathematically the <b>inspection paradox</b> states: <i>for any t > 0 the renewal interval containing t is <a href="/wiki/Stochastically_larger" class="mw-redirect" title="Stochastically larger">stochastically larger</a> than the first renewal interval.</i> That is, for all <i>x</i> > 0 and for all <i>t</i> > 0: </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 \operatorname {P} (S_{X_{t}+1}>x)\geq \operatorname {P} (S_{1}>x)=1-F_{S}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>></mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>></mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {P} (S_{X_{t}+1}>x)\geq \operatorname {P} (S_{1}>x)=1-F_{S}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/853256ad409cd08902895e87d55994be968dce6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.01ex; height:2.843ex;" alt="{\displaystyle \operatorname {P} (S_{X_{t}+1}>x)\geq \operatorname {P} (S_{1}>x)=1-F_{S}(x)}"></span></dd></dl> <p>where <i>F</i><sub><i>S</i></sub> is the cumulative distribution function of the IID holding times <i>S<sub>i</sub></i>. A vivid example is the <b>bus waiting time paradox</b>: For a given random distribution of bus arrivals, the average rider at a bus stop observes more delays than the average operator of the buses. </p><p>The resolution of the paradox is that our sampled distribution at time <i>t</i> is size-biased (see <a href="/wiki/Sampling_bias" title="Sampling bias">sampling bias</a>), in that the likelihood an interval is chosen is proportional to its size. However, a renewal interval of average size is not size-biased. </p> <dl><dd><table class="toccolours collapsible collapsed" width="80%" style="text-align:left"> <tbody><tr> <th>Proof </th></tr> <tr> <td>Observe that the last jump-time before <i>t</i> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{X_{t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{X_{t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7f47006853f6ae682855faf033eaab3fe21d44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.529ex; height:2.676ex;" alt="{\displaystyle J_{X_{t}}}"></span>; and that the renewal interval containing <i>t</i> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{X_{t}+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{X_{t}+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e1f3d34d55b484f37e195bbbdd7e90e6db8b507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.765ex; height:2.676ex;" alt="{\displaystyle S_{X_{t}+1}}"></span>. Then <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 {\begin{aligned}\operatorname {P} (S_{X_{t}+1}>x)&{}=\int _{0}^{\infty }\operatorname {P} (S_{X_{t}+1}>x\mid J_{X_{t}}=s)f_{J_{X_{t}}}(s)\,ds\\[12pt]&{}=\int _{0}^{\infty }\operatorname {P} (S_{X_{t}+1}>x|S_{X_{t}+1}>t-s)f_{J_{X_{t}}}(s)\,ds\\[12pt]&{}=\int _{0}^{\infty }{\frac {\operatorname {P} (S_{X_{t}+1}>x\,,\,S_{X_{t}+1}>t-s)}{\operatorname {P} (S_{X_{t}+1}>t-s)}}f_{J_{X_{t}}}(s)\,ds\\[12pt]&{}=\int _{0}^{\infty }{\frac {1-F(\max\{x,t-s\})}{1-F(t-s)}}f_{J_{X_{t}}}(s)\,ds\\[12pt]&{}=\int _{0}^{\infty }\min \left\{{\frac {1-F(x)}{1-F(t-s)}},{\frac {1-F(t-s)}{1-F(t-s)}}\right\}f_{J_{X_{t}}}(s)\,ds\\[12pt]&{}=\int _{0}^{\infty }\min \left\{{\frac {1-F(x)}{1-F(t-s)}},1\right\}f_{J_{X_{t}}}(s)\,ds\\[12pt]&{}\geq \int _{0}^{\infty }(1-F(x))f_{J_{X_{t}}}(s)\,ds=1-F(x)=\operatorname {P} (S_{1}>x),\\[12pt]\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.5em 1.5em 1.5em 1.5em 1.5em 1.5em 1.5em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>></mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>></mo> <mi>x</mi> <mo>∣</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mi>s</mi> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>></mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>></mo> <mi>t</mi> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>></mo> <mi>x</mi> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>></mo> <mi>t</mi> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>></mo> <mi>t</mi> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>−</mo> <mi>s</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo movablelimits="true" form="prefix">min</mo> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mrow> <mo>}</mo> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo movablelimits="true" form="prefix">min</mo> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>≥</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>></mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {P} (S_{X_{t}+1}>x)&{}=\int _{0}^{\infty }\operatorname {P} (S_{X_{t}+1}>x\mid J_{X_{t}}=s)f_{J_{X_{t}}}(s)\,ds\\[12pt]&{}=\int _{0}^{\infty }\operatorname {P} (S_{X_{t}+1}>x|S_{X_{t}+1}>t-s)f_{J_{X_{t}}}(s)\,ds\\[12pt]&{}=\int _{0}^{\infty }{\frac {\operatorname {P} (S_{X_{t}+1}>x\,,\,S_{X_{t}+1}>t-s)}{\operatorname {P} (S_{X_{t}+1}>t-s)}}f_{J_{X_{t}}}(s)\,ds\\[12pt]&{}=\int _{0}^{\infty }{\frac {1-F(\max\{x,t-s\})}{1-F(t-s)}}f_{J_{X_{t}}}(s)\,ds\\[12pt]&{}=\int _{0}^{\infty }\min \left\{{\frac {1-F(x)}{1-F(t-s)}},{\frac {1-F(t-s)}{1-F(t-s)}}\right\}f_{J_{X_{t}}}(s)\,ds\\[12pt]&{}=\int _{0}^{\infty }\min \left\{{\frac {1-F(x)}{1-F(t-s)}},1\right\}f_{J_{X_{t}}}(s)\,ds\\[12pt]&{}\geq \int _{0}^{\infty }(1-F(x))f_{J_{X_{t}}}(s)\,ds=1-F(x)=\operatorname {P} (S_{1}>x),\\[12pt]\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca6a13a416b0b4ef32dc077c4bb052983c507097" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -30.005ex; margin-top: -0.204ex; width:68.253ex; height:61.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {P} (S_{X_{t}+1}>x)&{}=\int _{0}^{\infty }\operatorname {P} (S_{X_{t}+1}>x\mid J_{X_{t}}=s)f_{J_{X_{t}}}(s)\,ds\\[12pt]&{}=\int _{0}^{\infty }\operatorname {P} (S_{X_{t}+1}>x|S_{X_{t}+1}>t-s)f_{J_{X_{t}}}(s)\,ds\\[12pt]&{}=\int _{0}^{\infty }{\frac {\operatorname {P} (S_{X_{t}+1}>x\,,\,S_{X_{t}+1}>t-s)}{\operatorname {P} (S_{X_{t}+1}>t-s)}}f_{J_{X_{t}}}(s)\,ds\\[12pt]&{}=\int _{0}^{\infty }{\frac {1-F(\max\{x,t-s\})}{1-F(t-s)}}f_{J_{X_{t}}}(s)\,ds\\[12pt]&{}=\int _{0}^{\infty }\min \left\{{\frac {1-F(x)}{1-F(t-s)}},{\frac {1-F(t-s)}{1-F(t-s)}}\right\}f_{J_{X_{t}}}(s)\,ds\\[12pt]&{}=\int _{0}^{\infty }\min \left\{{\frac {1-F(x)}{1-F(t-s)}},1\right\}f_{J_{X_{t}}}(s)\,ds\\[12pt]&{}\geq \int _{0}^{\infty }(1-F(x))f_{J_{X_{t}}}(s)\,ds=1-F(x)=\operatorname {P} (S_{1}>x),\\[12pt]\end{aligned}}}"></span></dd></dl> <p>since both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1-F(x)}{1-F(t-s)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1-F(x)}{1-F(t-s)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db65e03a559f7cd18b841c79e1144e907c508b7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.16ex; height:6.509ex;" alt="{\displaystyle {\frac {1-F(x)}{1-F(t-s)}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> are greater than or equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-F(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-F(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/635e431e0ebae6fae587c49c477945a831d387e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.883ex; height:2.843ex;" alt="{\displaystyle 1-F(x)}"></span> for all values of <i>s</i>. </p> </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Superposition">Superposition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Renewal_theory&action=edit&section=15" title="Edit section: Superposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Unless the renewal process is a Poisson process, the superposition (sum) of two independent renewal processes is not a renewal process.<sup id="cite_ref-FOOTNOTEGrimmettStirzaker1992405_7-0" class="reference"><a href="#cite_note-FOOTNOTEGrimmettStirzaker1992405-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> However, such processes can be described within a larger class of processes called the <a href="/wiki/Markov_renewal_process" title="Markov renewal process">Markov-renewal processes</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> However, the <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> of the first inter-event time in the superposition process is given by<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <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 R(t)=1-\sum _{k=1}^{K}{\frac {\alpha _{k}}{\sum _{l=1}^{K}\alpha _{l}}}(1-R_{k}(t))\prod _{j=1,j\neq k}^{K}\alpha _{j}\int _{t}^{\infty }(1-R_{j}(u))\,{\text{d}}u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>≠</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(t)=1-\sum _{k=1}^{K}{\frac {\alpha _{k}}{\sum _{l=1}^{K}\alpha _{l}}}(1-R_{k}(t))\prod _{j=1,j\neq k}^{K}\alpha _{j}\int _{t}^{\infty }(1-R_{j}(u))\,{\text{d}}u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a52b58f1f0a020c6fa1f32f44010d8f442a60f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:63.86ex; height:7.843ex;" alt="{\displaystyle R(t)=1-\sum _{k=1}^{K}{\frac {\alpha _{k}}{\sum _{l=1}^{K}\alpha _{l}}}(1-R_{k}(t))\prod _{j=1,j\neq k}^{K}\alpha _{j}\int _{t}^{\infty }(1-R_{j}(u))\,{\text{d}}u}"></span></dd></dl> <p>where <i>R</i><sub><i>k</i></sub>(<i>t</i>) and <i>α</i><sub><i>k</i></sub> > 0 are the CDF of the inter-event times and the arrival rate of process <i>k</i>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Example_application">Example application</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Renewal_theory&action=edit&section=16" title="Edit section: Example application"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Eric the entrepreneur has <i>n</i> machines, each having an operational lifetime uniformly distributed between zero and two years. Eric may let each machine run until it fails with replacement cost €2600; alternatively he may replace a machine at any time while it is still functional at a cost of €200. </p><p>What is his optimal replacement policy? </p> <dl><dd><table class="toccolours collapsible collapsed" width="80%" style="text-align:left"> <tbody><tr> <th>Solution </th></tr> <tr> <td>The lifetime of the <i>n</i> machines can be modeled as <i>n</i> independent concurrent renewal-reward processes, so it is sufficient to consider the case <i>n=1</i>. Denote this process by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Y_{t})_{t\geq 0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <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> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Y_{t})_{t\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a52866b4fb8967514ea2b18a95a6a06348bc368" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.912ex; height:2.843ex;" alt="{\displaystyle (Y_{t})_{t\geq 0}}"></span>. The successive lifetimes <i>S</i> of the replacement machines are independent and identically distributed, so the optimal policy is the same for all replacement machines in the process. <p>If Eric decides at the start of a machine's life to replace it at time <span class="nowrap">0 < <i>t</i> < 2</span> but the machine happens to fail before that time then the lifetime <i>S</i> of the machine is uniformly distributed on [0, <i>t</i>] and thus has expectation 0.5<i>t</i>. So the overall expected lifetime of the machine is: </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 {\begin{aligned}\operatorname {E} [S]&=\operatorname {E} [S\mid {\text{fails before }}t]\cdot \operatorname {P} [{\text{fails before }}t]+\operatorname {E} [S\mid {\text{does not fail before }}t]\cdot \operatorname {P} [{\text{does not fail before }}t]\\[6pt]&=0.5t({\frac {t}{2}})+t({\frac {2-t}{2}})\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>S</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>S</mi> <mo>∣</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>fails before </mtext> </mrow> <mi>t</mi> <mo stretchy="false">]</mo> <mo>⋅</mo> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>fails before </mtext> </mrow> <mi>t</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>S</mi> <mo>∣</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>does not fail before </mtext> </mrow> <mi>t</mi> <mo stretchy="false">]</mo> <mo>⋅</mo> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>does not fail before </mtext> </mrow> <mi>t</mi> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>0.5</mn> <mi>t</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>t</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo>−</mo> <mi>t</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} [S]&=\operatorname {E} [S\mid {\text{fails before }}t]\cdot \operatorname {P} [{\text{fails before }}t]+\operatorname {E} [S\mid {\text{does not fail before }}t]\cdot \operatorname {P} [{\text{does not fail before }}t]\\[6pt]&=0.5t({\frac {t}{2}})+t({\frac {2-t}{2}})\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8e0739e112060763178a0e31b789d8100fca8e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:97.733ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}\operatorname {E} [S]&=\operatorname {E} [S\mid {\text{fails before }}t]\cdot \operatorname {P} [{\text{fails before }}t]+\operatorname {E} [S\mid {\text{does not fail before }}t]\cdot \operatorname {P} [{\text{does not fail before }}t]\\[6pt]&=0.5t({\frac {t}{2}})+t({\frac {2-t}{2}})\end{aligned}}}"></span></dd></dl> <p>and the expected cost <i>W</i> per machine is: </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 {\begin{aligned}\operatorname {E} [W]&=\operatorname {E} [W\mid {\text{fails before }}t]\cdot \operatorname {P} ({\text{fails before }}t)+\operatorname {E} [W\mid {\text{does not fail before }}t]\cdot \operatorname {P} ({\text{does not fail before }}t)\\[6pt]&=2600({\frac {t}{2}})+200({\frac {2-t}{2}})=1200t+200.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>W</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>W</mi> <mo>∣</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>fails before </mtext> </mrow> <mi>t</mi> <mo stretchy="false">]</mo> <mo>⋅</mo> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>fails before </mtext> </mrow> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>W</mi> <mo>∣</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>does not fail before </mtext> </mrow> <mi>t</mi> <mo stretchy="false">]</mo> <mo>⋅</mo> <mi mathvariant="normal">P</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>does not fail before </mtext> </mrow> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2600</mn> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mn>200</mn> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo>−</mo> <mi>t</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1200</mn> <mi>t</mi> <mo>+</mo> <mn>200.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} [W]&=\operatorname {E} [W\mid {\text{fails before }}t]\cdot \operatorname {P} ({\text{fails before }}t)+\operatorname {E} [W\mid {\text{does not fail before }}t]\cdot \operatorname {P} ({\text{does not fail before }}t)\\[6pt]&=2600({\frac {t}{2}})+200({\frac {2-t}{2}})=1200t+200.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/085216b522dfbf33cb3fc71290697b12503962a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:101.573ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}\operatorname {E} [W]&=\operatorname {E} [W\mid {\text{fails before }}t]\cdot \operatorname {P} ({\text{fails before }}t)+\operatorname {E} [W\mid {\text{does not fail before }}t]\cdot \operatorname {P} ({\text{does not fail before }}t)\\[6pt]&=2600({\frac {t}{2}})+200({\frac {2-t}{2}})=1200t+200.\end{aligned}}}"></span></dd></dl> <p>So by the strong law of large numbers, his long-term average cost per unit time is: </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 {\frac {1}{t}}Y_{t}\simeq {\frac {\operatorname {E} [W]}{\operatorname {E} [S]}}={\frac {4(1200t+200)}{t^{2}+4t-2t^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>t</mi> </mfrac> </mrow> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>≃</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>W</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>S</mi> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mo stretchy="false">(</mo> <mn>1200</mn> <mi>t</mi> <mo>+</mo> <mn>200</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>t</mi> <mo>−</mo> <mn>2</mn> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{t}}Y_{t}\simeq {\frac {\operatorname {E} [W]}{\operatorname {E} [S]}}={\frac {4(1200t+200)}{t^{2}+4t-2t^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1fb2305b9678f98f546ae389c2f277163434afd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.145ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{t}}Y_{t}\simeq {\frac {\operatorname {E} [W]}{\operatorname {E} [S]}}={\frac {4(1200t+200)}{t^{2}+4t-2t^{2}}}}"></span></dd></dl> <p>then differentiating with respect to <i>t</i>: </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 {\frac {\partial }{\partial t}}{\frac {4(1200t+200)}{t^{2}+4t-2t^{2}}}=4{\frac {(4t-t^{2})(1200)-(4-2t)(1200t+200)}{(t^{2}+4t-2t^{2})^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mo stretchy="false">(</mo> <mn>1200</mn> <mi>t</mi> <mo>+</mo> <mn>200</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>t</mi> <mo>−</mo> <mn>2</mn> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mi>t</mi> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1200</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo>−</mo> <mn>2</mn> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1200</mn> <mi>t</mi> <mo>+</mo> <mn>200</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>t</mi> <mo>−</mo> <mn>2</mn> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial t}}{\frac {4(1200t+200)}{t^{2}+4t-2t^{2}}}=4{\frac {(4t-t^{2})(1200)-(4-2t)(1200t+200)}{(t^{2}+4t-2t^{2})^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c09c52b81cf2c7d50750f67d1c98134d39990f69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:63.649ex; height:6.676ex;" alt="{\displaystyle {\frac {\partial }{\partial t}}{\frac {4(1200t+200)}{t^{2}+4t-2t^{2}}}=4{\frac {(4t-t^{2})(1200)-(4-2t)(1200t+200)}{(t^{2}+4t-2t^{2})^{2}}},}"></span></dd></dl> <p>this implies that the turning points satisfy: </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 {\begin{aligned}0&=(4t-t^{2})(1200)-(4-2t)(1200t+200)=4800t-1200t^{2}-4800t-800+2400t^{2}+400t\\[6pt]&=-800+400t+1200t^{2},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>t</mi> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1200</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo>−</mo> <mn>2</mn> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1200</mn> <mi>t</mi> <mo>+</mo> <mn>200</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4800</mn> <mi>t</mi> <mo>−</mo> <mn>1200</mn> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4800</mn> <mi>t</mi> <mo>−</mo> <mn>800</mn> <mo>+</mo> <mn>2400</mn> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>400</mn> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>800</mn> <mo>+</mo> <mn>400</mn> <mi>t</mi> <mo>+</mo> <mn>1200</mn> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}0&=(4t-t^{2})(1200)-(4-2t)(1200t+200)=4800t-1200t^{2}-4800t-800+2400t^{2}+400t\\[6pt]&=-800+400t+1200t^{2},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79ab7f68475f7573e4e243232e03af08b09d5845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:93.479ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}0&=(4t-t^{2})(1200)-(4-2t)(1200t+200)=4800t-1200t^{2}-4800t-800+2400t^{2}+400t\\[6pt]&=-800+400t+1200t^{2},\end{aligned}}}"></span></dd></dl> <p>and thus </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 0=3t^{2}+t-2=(3t-2)(t+1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mn>3</mn> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>t</mi> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>t</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=3t^{2}+t-2=(3t-2)(t+1).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/957ec4158cd8d37572268c29c813481360bec856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.211ex; height:3.176ex;" alt="{\displaystyle 0=3t^{2}+t-2=(3t-2)(t+1).}"></span></dd></dl> <p>We take the only solution <i>t</i> in [0, 2]: <i>t</i> = 2/3. This is indeed a minimum (and not a maximum) since the cost per unit time tends to infinity as <i>t</i> tends to zero, meaning that the cost is decreasing as <i>t</i> increases, until the point 2/3 where it starts to increase. </p> </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Renewal_theory&action=edit&section=17" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 30em;"> <ul><li><a href="/wiki/Campbell%27s_theorem_(probability)" title="Campbell's theorem (probability)">Campbell's theorem (probability)</a></li> <li><a href="/wiki/Compound_Poisson_process" title="Compound Poisson process">Compound Poisson process</a></li> <li><a href="/wiki/Continuous-time_Markov_process" class="mw-redirect" title="Continuous-time Markov process">Continuous-time Markov process</a></li> <li><a href="/wiki/Little%27s_lemma" class="mw-redirect" title="Little's lemma">Little's lemma</a></li> <li><a href="/wiki/Lotka%27s_integral_equation" class="mw-redirect" title="Lotka's integral equation">Lotka's integral equation</a></li> <li><a href="/wiki/Palm%E2%80%93Khintchine_theorem" title="Palm–Khintchine theorem">Palm–Khintchine theorem</a></li> <li><a href="/wiki/Poisson_process" class="mw-redirect" title="Poisson process">Poisson process</a></li> <li><a href="/wiki/Queueing_theory" title="Queueing theory">Queueing theory</a></li> <li><a href="/wiki/Residual_time" title="Residual time">Residual time</a></li> <li><a href="/wiki/Ruin_theory" title="Ruin theory">Ruin theory</a></li> <li><a href="/wiki/Semi-Markov_process" class="mw-redirect" title="Semi-Markov process">Semi-Markov process</a></li> <li><a href="/wiki/Von_Foerster_equation" title="Von Foerster equation">Von Foerster equation</a></li></ul> </div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output 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.reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-FOOTNOTEGrimmettStirzaker1992393-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGrimmettStirzaker1992393_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGrimmettStirzaker1992">Grimmett & Stirzaker (1992)</a>, p. 393.</span> </li> <li id="cite_note-FOOTNOTEGrimmettStirzaker1992390-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGrimmettStirzaker1992390_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGrimmettStirzaker1992">Grimmett & Stirzaker (1992)</a>, p. 390.</span> </li> <li id="cite_note-FOOTNOTEGrimmettStirzaker1992395-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEGrimmettStirzaker1992395_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEGrimmettStirzaker1992395_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTEGrimmettStirzaker1992395_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFGrimmettStirzaker1992">Grimmett & Stirzaker (1992)</a>, p. 395.</span> </li> <li id="cite_note-FOOTNOTEFeller1971347–351-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFeller1971347–351_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFeller1971">Feller (1971)</a>, p. 347–351.</span> </li> <li id="cite_note-FOOTNOTEGrimmettStirzaker1992394–5-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGrimmettStirzaker1992394–5_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGrimmettStirzaker1992">Grimmett & Stirzaker (1992)</a>, p. 394–5.</span> </li> <li id="cite_note-FOOTNOTEGrimmettStirzaker1992394-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEGrimmettStirzaker1992394_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEGrimmettStirzaker1992394_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFGrimmettStirzaker1992">Grimmett & Stirzaker (1992)</a>, p. 394.</span> </li> <li id="cite_note-FOOTNOTEGrimmettStirzaker1992405-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGrimmettStirzaker1992405_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGrimmettStirzaker1992">Grimmett & Stirzaker (1992)</a>, p. 405.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFÇinlar1969" class="citation journal cs1"><a href="/wiki/Erhan_Cinlar" class="mw-redirect" title="Erhan Cinlar">Çinlar, Erhan</a> (1969). "Markov Renewal Theory". <i>Advances in Applied Probability</i>. <b>1</b> (2). Applied Probability Trust: 123–187. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1426216">10.2307/1426216</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1426216">1426216</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advances+in+Applied+Probability&rft.atitle=Markov+Renewal+Theory&rft.volume=1&rft.issue=2&rft.pages=123-187&rft.date=1969&rft_id=info%3Adoi%2F10.2307%2F1426216&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1426216%23id-name%3DJSTOR&rft.aulast=%C3%87inlar&rft.aufirst=Erhan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARenewal+theory" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLawrence1973" class="citation journal cs1">Lawrence, A. J. (1973). "Dependency of Intervals Between Events in Superposition Processes". <i>Journal of the Royal Statistical Society. Series B (Methodological)</i>. <b>35</b> (2): 306–315. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.2517-6161.1973.tb00960.x">10.1111/j.2517-6161.1973.tb00960.x</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2984914">2984914</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Royal+Statistical+Society.+Series+B+%28Methodological%29&rft.atitle=Dependency+of+Intervals+Between+Events+in+Superposition+Processes&rft.volume=35&rft.issue=2&rft.pages=306-315&rft.date=1973&rft_id=info%3Adoi%2F10.1111%2Fj.2517-6161.1973.tb00960.x&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2984914%23id-name%3DJSTOR&rft.aulast=Lawrence&rft.aufirst=A.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARenewal+theory" class="Z3988"></span> formula 4.1</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChoungmo_FofackNainNegliaTowsley2012" class="citation thesis cs1">Choungmo Fofack, Nicaise; Nain, Philippe; Neglia, Giovanni; <a href="/wiki/Don_Towsley_(computer_scientist)" title="Don Towsley (computer scientist)">Towsley, Don</a> (6 March 2012). <a rel="nofollow" class="external text" href="http://hal.inria.fr/hal-00676735"><i>Analysis of TTL-based Cache Networks</i></a>. <i>Proceedings of 6th International Conference on Performance Evaluation Methodologies and Tools</i> (report)<span class="reference-accessdate">. Retrieved <span class="nowrap">Nov 15,</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=Analysis+of+TTL-based+Cache+Networks&rft.date=2012-03-06&rft.aulast=Choungmo+Fofack&rft.aufirst=Nicaise&rft.au=Nain%2C+Philippe&rft.au=Neglia%2C+Giovanni&rft.au=Towsley%2C+Don&rft_id=http%3A%2F%2Fhal.inria.fr%2Fhal-00676735&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARenewal+theory" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Renewal_theory&action=edit&section=19" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="cox" class="citation book cs1"><a href="/wiki/Sir_David_Cox_(statistician)" class="mw-redirect" title="Sir David Cox (statistician)">Cox, David</a> (1970). <i>Renewal Theory</i>. London: Methuen & Co. p. 142. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-412-20570-X" title="Special:BookSources/0-412-20570-X"><bdi>0-412-20570-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Renewal+Theory&rft.place=London&rft.pages=142&rft.pub=Methuen+%26+Co.&rft.date=1970&rft.isbn=0-412-20570-X&rft.aulast=Cox&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARenewal+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDoob1948" class="citation journal cs1">Doob, J. L. (1948). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/tran/1948-063-03/S0002-9947-1948-0025098-8/S0002-9947-1948-0025098-8.pdf">"Renewal Theory From the Point of View of the Theory of Probability"</a> <span class="cs1-format">(PDF)</span>. <i>Transactions of the American Mathematical Society</i>. <b>63</b> (3): 422–438. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1990567">10.2307/1990567</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1990567">1990567</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=Renewal+Theory+From+the+Point+of+View+of+the+Theory+of+Probability&rft.volume=63&rft.issue=3&rft.pages=422-438&rft.date=1948&rft_id=info%3Adoi%2F10.2307%2F1990567&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1990567%23id-name%3DJSTOR&rft.aulast=Doob&rft.aufirst=J.+L.&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Ftran%2F1948-063-03%2FS0002-9947-1948-0025098-8%2FS0002-9947-1948-0025098-8.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARenewal+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeller1971" class="citation book cs1"><a href="/wiki/William_Feller" title="William Feller">Feller, William</a> (1971). <i>An introduction to probability theory and its applications</i>. Vol. 2 (second ed.). Wiley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+probability+theory+and+its+applications&rft.edition=second&rft.pub=Wiley&rft.date=1971&rft.aulast=Feller&rft.aufirst=William&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARenewal+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrimmettStirzaker1992" class="citation book cs1"><a href="/wiki/Geoffrey_Grimmett" title="Geoffrey Grimmett">Grimmett, G. R.</a>; Stirzaker, D. R. (1992). <i>Probability and Random Processes</i> (second ed.). Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0198572220" title="Special:BookSources/0198572220"><bdi>0198572220</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability+and+Random+Processes&rft.edition=second&rft.pub=Oxford+University+Press&rft.date=1992&rft.isbn=0198572220&rft.aulast=Grimmett&rft.aufirst=G.+R.&rft.au=Stirzaker%2C+D.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARenewal+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith1958" class="citation journal cs1">Smith, Walter L. (1958). "Renewal Theory and Its Ramifications". <i><a href="/wiki/Journal_of_the_Royal_Statistical_Society,_Series_B" class="mw-redirect" title="Journal of the Royal Statistical Society, Series B">Journal of the Royal Statistical Society, Series B</a></i>. <b>20</b> (2): 243–302. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2983891">2983891</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Royal+Statistical+Society%2C+Series+B&rft.atitle=Renewal+Theory+and+Its+Ramifications&rft.volume=20&rft.issue=2&rft.pages=243-302&rft.date=1958&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2983891%23id-name%3DJSTOR&rft.aulast=Smith&rft.aufirst=Walter+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARenewal+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWanli_Wang,_Johannes_H._P._Schulz2018" class="citation journal cs1">Wanli Wang, Johannes H. P. Schulz, Weihua Deng, and Eli Barkai (2018). "Renewal theory with fat-tailed distributed sojourn times: Typical versus rare". <i>Phys. Rev. E</i>. <b>98</b> (4): 042139. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1809.05856">1809.05856</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2018PhRvE..98d2139W">2018PhRvE..98d2139W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevE.98.042139">10.1103/PhysRevE.98.042139</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:54727926">54727926</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phys.+Rev.+E&rft.atitle=Renewal+theory+with+fat-tailed+distributed+sojourn+times%3A+Typical+versus+rare&rft.volume=98&rft.issue=4&rft.pages=042139&rft.date=2018&rft_id=info%3Aarxiv%2F1809.05856&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A54727926%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevE.98.042139&rft_id=info%3Abibcode%2F2018PhRvE..98d2139W&rft.aulast=Wanli+Wang%2C+Johannes+H.+P.+Schulz&rft.aufirst=Weihua+Deng%2C+and+Eli+Barkai&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARenewal+theory" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></li></ul> <div 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scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete-time_stochastic_process" class="mw-redirect" title="Discrete-time stochastic process">Discrete time</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li> <li><a href="/wiki/Branching_process" title="Branching process">Branching process</a></li> <li><a href="/wiki/Chinese_restaurant_process" title="Chinese restaurant process">Chinese restaurant process</a></li> <li><a href="/wiki/Galton%E2%80%93Watson_process" title="Galton–Watson process">Galton–Watson process</a></li> <li><a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">Independent and identically distributed random variables</a></li> <li><a href="/wiki/Markov_chain" title="Markov chain">Markov chain</a></li> <li><a href="/wiki/Moran_process" title="Moran process">Moran process</a></li> <li><a href="/wiki/Random_walk" title="Random walk">Random walk</a> <ul><li><a href="/wiki/Loop-erased_random_walk" title="Loop-erased random walk">Loop-erased</a></li> <li><a href="/wiki/Self-avoiding_walk" title="Self-avoiding walk">Self-avoiding</a></li> <li><a href="/wiki/Biased_random_walk_on_a_graph" title="Biased random walk on a graph"> Biased</a></li> <li><a href="/wiki/Maximal_entropy_random_walk" title="Maximal entropy random walk">Maximal entropy</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Continuous-time_stochastic_process" title="Continuous-time stochastic process">Continuous time</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Additive_process" title="Additive process">Additive process</a></li> <li><a href="/wiki/Bessel_process" title="Bessel process">Bessel process</a></li> <li><a href="/wiki/Birth%E2%80%93death_process" title="Birth–death process">Birth–death process</a> <ul><li><a href="/wiki/Birth_process" title="Birth process">pure birth</a></li></ul></li> <li><a href="/wiki/Wiener_process" title="Wiener process">Brownian motion</a> <ul><li><a href="/wiki/Brownian_bridge" title="Brownian bridge">Bridge</a></li> <li><a href="/wiki/Brownian_excursion" title="Brownian excursion">Excursion</a></li> <li><a href="/wiki/Fractional_Brownian_motion" title="Fractional Brownian motion">Fractional</a></li> <li><a href="/wiki/Geometric_Brownian_motion" title="Geometric Brownian motion">Geometric</a></li> <li><a href="/wiki/Brownian_meander" title="Brownian meander">Meander</a></li></ul></li> <li><a href="/wiki/Cauchy_process" title="Cauchy process">Cauchy process</a></li> <li><a href="/wiki/Contact_process_(mathematics)" title="Contact process (mathematics)">Contact process</a></li> <li><a href="/wiki/Continuous-time_random_walk" title="Continuous-time random walk">Continuous-time random walk</a></li> <li><a href="/wiki/Cox_process" title="Cox process">Cox process</a></li> <li><a href="/wiki/Diffusion_process" title="Diffusion process">Diffusion process</a></li> <li><a href="/wiki/Dyson_Brownian_motion" title="Dyson Brownian motion">Dyson Brownian motion</a></li> <li><a href="/wiki/Empirical_process" title="Empirical process">Empirical process</a></li> <li><a href="/wiki/Feller_process" title="Feller process">Feller process</a></li> <li><a href="/wiki/Fleming%E2%80%93Viot_process" title="Fleming–Viot process">Fleming–Viot process</a></li> <li><a href="/wiki/Gamma_process" title="Gamma process">Gamma process</a></li> <li><a href="/wiki/Geometric_process" title="Geometric process">Geometric process</a></li> <li><a href="/wiki/Hawkes_process" title="Hawkes process">Hawkes process</a></li> <li><a href="/wiki/Hunt_process" title="Hunt process">Hunt process</a></li> <li><a href="/wiki/Interacting_particle_system" title="Interacting particle system">Interacting particle systems</a></li> <li><a href="/wiki/It%C3%B4_diffusion" title="Itô diffusion">Itô diffusion</a></li> <li><a href="/wiki/It%C3%B4_process" class="mw-redirect" title="Itô process">Itô process</a></li> <li><a href="/wiki/Jump_diffusion" title="Jump diffusion">Jump diffusion</a></li> <li><a href="/wiki/Jump_process" title="Jump process">Jump process</a></li> <li><a href="/wiki/L%C3%A9vy_process" title="Lévy process">Lévy process</a></li> <li><a href="/wiki/Local_time_(mathematics)" title="Local time (mathematics)">Local time</a></li> <li><a href="/wiki/Markov_additive_process" title="Markov additive process">Markov additive process</a></li> <li><a href="/wiki/McKean%E2%80%93Vlasov_process" title="McKean–Vlasov process">McKean–Vlasov process</a></li> <li><a href="/wiki/Ornstein%E2%80%93Uhlenbeck_process" title="Ornstein–Uhlenbeck process">Ornstein–Uhlenbeck process</a></li> <li><a href="/wiki/Poisson_point_process" title="Poisson point process">Poisson process</a> <ul><li><a href="/wiki/Compound_Poisson_process" title="Compound Poisson process">Compound</a></li> <li><a href="/wiki/Non-homogeneous_Poisson_process" class="mw-redirect" title="Non-homogeneous Poisson process">Non-homogeneous</a></li></ul></li> <li><a href="/wiki/Schramm%E2%80%93Loewner_evolution" title="Schramm–Loewner evolution">Schramm–Loewner evolution</a></li> <li><a href="/wiki/Semimartingale" title="Semimartingale">Semimartingale</a></li> <li><a href="/wiki/Sigma-martingale" title="Sigma-martingale">Sigma-martingale</a></li> <li><a href="/wiki/Stable_process" title="Stable process">Stable process</a></li> <li><a href="/wiki/Superprocess" title="Superprocess">Superprocess</a></li> <li><a href="/wiki/Telegraph_process" title="Telegraph process">Telegraph process</a></li> <li><a href="/wiki/Variance_gamma_process" title="Variance gamma process">Variance gamma process</a></li> <li><a href="/wiki/Wiener_process" title="Wiener process">Wiener process</a></li> <li><a href="/wiki/Wiener_sausage" title="Wiener sausage">Wiener sausage</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Both</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Branching_process" title="Branching process">Branching process</a></li> <li><a href="/wiki/Gaussian_process" title="Gaussian process">Gaussian process</a></li> <li><a href="/wiki/Hidden_Markov_model" title="Hidden Markov model">Hidden Markov model (HMM)</a></li> <li><a href="/wiki/Markov_process" class="mw-redirect" title="Markov process">Markov process</a></li> <li><a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">Martingale</a> <ul><li><a href="/wiki/Martingale_difference_sequence" title="Martingale difference sequence">Differences</a></li> <li><a href="/wiki/Local_martingale" title="Local martingale">Local</a></li> <li><a href="/wiki/Submartingale" class="mw-redirect" title="Submartingale">Sub-</a></li> <li><a href="/wiki/Supermartingale" class="mw-redirect" title="Supermartingale">Super-</a></li></ul></li> <li><a href="/wiki/Random_dynamical_system" title="Random dynamical system">Random dynamical system</a></li> <li><a href="/wiki/Regenerative_process" title="Regenerative process">Regenerative process</a></li> <li><a href="/wiki/Renewal_process" class="mw-redirect" title="Renewal process">Renewal process</a></li> <li><a href="/wiki/Stochastic_chains_with_memory_of_variable_length" title="Stochastic chains with memory of variable length">Stochastic chains with memory of variable length</a></li> <li><a href="/wiki/White_noise" title="White noise">White noise</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fields and other</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dirichlet_process" title="Dirichlet process">Dirichlet process</a></li> <li><a href="/wiki/Gaussian_random_field" title="Gaussian random field">Gaussian random field</a></li> <li><a href="/wiki/Gibbs_measure" title="Gibbs measure">Gibbs measure</a></li> <li><a href="/wiki/Hopfield_model" class="mw-redirect" title="Hopfield model">Hopfield model</a></li> <li><a href="/wiki/Ising_model" title="Ising model">Ising model</a> <ul><li><a href="/wiki/Potts_model" title="Potts model">Potts model</a></li> <li><a href="/wiki/Boolean_network" title="Boolean network">Boolean network</a></li></ul></li> <li><a href="/wiki/Markov_random_field" title="Markov random field">Markov random field</a></li> <li><a href="/wiki/Percolation_theory" title="Percolation theory">Percolation</a></li> <li><a href="/wiki/Pitman%E2%80%93Yor_process" title="Pitman–Yor process">Pitman–Yor process</a></li> <li><a href="/wiki/Point_process" title="Point process">Point process</a> <ul><li><a href="/wiki/Point_process#Cox_point_process" title="Point process">Cox</a></li> <li><a href="/wiki/Poisson_point_process" title="Poisson point process">Poisson</a></li></ul></li> <li><a href="/wiki/Random_field" title="Random field">Random field</a></li> <li><a href="/wiki/Random_graph" title="Random graph">Random graph</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Time_series" title="Time series">Time series models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH) model</a></li> <li><a href="/wiki/Autoregressive_integrated_moving_average" title="Autoregressive integrated moving average">Autoregressive integrated moving average (ARIMA) model</a></li> <li><a href="/wiki/Autoregressive_model" title="Autoregressive model">Autoregressive (AR) model</a></li> <li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">Autoregressive–moving-average (ARMA) model</a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Generalized autoregressive conditional heteroskedasticity (GARCH) model</a></li> <li><a href="/wiki/Moving-average_model" title="Moving-average model">Moving-average (MA) model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Asset_pricing_model" class="mw-redirect" title="Asset pricing model">Financial models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binomial_options_pricing_model" title="Binomial options pricing model">Binomial options pricing model</a></li> <li><a href="/wiki/Black%E2%80%93Derman%E2%80%93Toy_model" title="Black–Derman–Toy model">Black–Derman–Toy</a></li> <li><a href="/wiki/Black%E2%80%93Karasinski_model" title="Black–Karasinski model">Black–Karasinski</a></li> <li><a href="/wiki/Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes</a></li> <li><a href="/wiki/Chan%E2%80%93Karolyi%E2%80%93Longstaff%E2%80%93Sanders_process" title="Chan–Karolyi–Longstaff–Sanders process">Chan–Karolyi–Longstaff–Sanders (CKLS)</a></li> <li><a href="/wiki/Chen_model" title="Chen model">Chen</a></li> <li><a href="/wiki/Constant_elasticity_of_variance_model" title="Constant elasticity of variance model">Constant elasticity of variance (CEV)</a></li> <li><a href="/wiki/Cox%E2%80%93Ingersoll%E2%80%93Ross_model" title="Cox–Ingersoll–Ross model">Cox–Ingersoll–Ross (CIR)</a></li> <li><a href="/wiki/Garman%E2%80%93Kohlhagen_model" class="mw-redirect" title="Garman–Kohlhagen model">Garman–Kohlhagen</a></li> <li><a href="/wiki/Heath%E2%80%93Jarrow%E2%80%93Morton_framework" title="Heath–Jarrow–Morton framework">Heath–Jarrow–Morton (HJM)</a></li> <li><a href="/wiki/Heston_model" title="Heston model">Heston</a></li> <li><a href="/wiki/Ho%E2%80%93Lee_model" title="Ho–Lee model">Ho–Lee</a></li> <li><a href="/wiki/Hull%E2%80%93White_model" title="Hull–White model">Hull–White</a></li> <li><a href="/wiki/Korn%E2%80%93Kreer%E2%80%93Lenssen_model" title="Korn–Kreer–Lenssen model">Korn-Kreer-Lenssen</a></li> <li><a href="/wiki/LIBOR_market_model" title="LIBOR market model">LIBOR market</a></li> <li><a href="/wiki/Rendleman%E2%80%93Bartter_model" title="Rendleman–Bartter model">Rendleman–Bartter</a></li> <li><a href="/wiki/SABR_volatility_model" title="SABR volatility model">SABR volatility</a></li> <li><a href="/wiki/Vasicek_model" title="Vasicek model">Vašíček</a></li> <li><a href="/wiki/Wilkie_investment_model" title="Wilkie investment model">Wilkie</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/B%C3%BChlmann_model" title="Bühlmann model">Bühlmann</a></li> <li><a href="/wiki/Cram%C3%A9r%E2%80%93Lundberg_model" class="mw-redirect" title="Cramér–Lundberg model">Cramér–Lundberg</a></li> <li><a href="/wiki/Risk_process" class="mw-redirect" title="Risk process">Risk process</a></li> <li><a href="/wiki/Sparre%E2%80%93Anderson_model" class="mw-redirect" title="Sparre–Anderson model">Sparre–Anderson</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Queueing_model" class="mw-redirect" title="Queueing model">Queueing models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bulk_queue" title="Bulk queue">Bulk</a></li> <li><a href="/wiki/Fluid_queue" title="Fluid queue">Fluid</a></li> <li><a href="/wiki/G-network" title="G-network">Generalized queueing network</a></li> <li><a href="/wiki/M/G/1_queue" title="M/G/1 queue">M/G/1</a></li> <li><a href="/wiki/M/M/1_queue" title="M/M/1 queue">M/M/1</a></li> <li><a href="/wiki/M/M/c_queue" title="M/M/c queue">M/M/c</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/C%C3%A0dl%C3%A0g" title="Càdlàg">Càdlàg paths</a></li> <li><a href="/wiki/Continuous_stochastic_process" title="Continuous stochastic process">Continuous</a></li> <li><a href="/wiki/Sample-continuous_process" title="Sample-continuous process">Continuous paths</a></li> <li><a href="/wiki/Ergodicity" title="Ergodicity">Ergodic</a></li> <li><a href="/wiki/Exchangeable_random_variables" title="Exchangeable random variables">Exchangeable</a></li> <li><a href="/wiki/Feller-continuous_process" title="Feller-continuous process">Feller-continuous</a></li> <li><a href="/wiki/Gauss%E2%80%93Markov_process" title="Gauss–Markov process">Gauss–Markov</a></li> <li><a href="/wiki/Markov_property" title="Markov property">Markov</a></li> <li><a href="/wiki/Mixing_(mathematics)" title="Mixing (mathematics)">Mixing</a></li> <li><a href="/wiki/Piecewise-deterministic_Markov_process" title="Piecewise-deterministic Markov process">Piecewise-deterministic</a></li> <li><a href="/wiki/Predictable_process" title="Predictable process">Predictable</a></li> <li><a href="/wiki/Progressively_measurable_process" title="Progressively measurable process">Progressively measurable</a></li> <li><a href="/wiki/Self-similar_process" title="Self-similar process">Self-similar</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationary</a></li> <li><a href="/wiki/Time_reversibility" title="Time reversibility">Time-reversible</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Limit theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li> <li><a href="/wiki/Donsker%27s_theorem" title="Donsker's theorem">Donsker's theorem</a></li> <li><a href="/wiki/Doob%27s_martingale_convergence_theorems" title="Doob's martingale convergence theorems">Doob's martingale convergence theorems</a></li> <li><a href="/wiki/Ergodic_theorem" class="mw-redirect" title="Ergodic theorem">Ergodic theorem</a></li> <li><a href="/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem" title="Fisher–Tippett–Gnedenko theorem">Fisher–Tippett–Gnedenko theorem</a></li> <li><a href="/wiki/Large_deviation_principle" class="mw-redirect" title="Large deviation principle">Large deviation principle</a></li> <li><a href="/wiki/Law_of_large_numbers" title="Law of large numbers">Law of large numbers (weak/strong)</a></li> <li><a href="/wiki/Law_of_the_iterated_logarithm" title="Law of the iterated logarithm">Law of the iterated logarithm</a></li> <li><a href="/wiki/Maximal_ergodic_theorem" title="Maximal ergodic theorem">Maximal ergodic theorem</a></li> <li><a href="/wiki/Sanov%27s_theorem" title="Sanov's theorem">Sanov's theorem</a></li> <li><a href="/wiki/Zero%E2%80%93one_law" title="Zero–one law">Zero–one laws</a> (<a href="/wiki/Blumenthal%27s_zero%E2%80%93one_law" title="Blumenthal's zero–one law">Blumenthal</a>, <a href="/wiki/Borel%E2%80%93Cantelli_lemma" title="Borel–Cantelli lemma">Borel–Cantelli</a>, <a href="/wiki/Engelbert%E2%80%93Schmidt_zero%E2%80%93one_law" title="Engelbert–Schmidt zero–one law">Engelbert–Schmidt</a>, <a href="/wiki/Hewitt%E2%80%93Savage_zero%E2%80%93one_law" title="Hewitt–Savage zero–one law">Hewitt–Savage</a>, <a href="/wiki/Kolmogorov%27s_zero%E2%80%93one_law" title="Kolmogorov's zero–one law"> Kolmogorov</a>, <a href="/wiki/L%C3%A9vy%27s_zero%E2%80%93one_law" class="mw-redirect" title="Lévy's zero–one law">Lévy</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_inequalities#Probability_theory_and_statistics" title="List of inequalities">Inequalities</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Burkholder%E2%80%93Davis%E2%80%93Gundy_inequalities" class="mw-redirect" title="Burkholder–Davis–Gundy inequalities">Burkholder–Davis–Gundy</a></li> <li><a href="/wiki/Doob%27s_martingale_inequality" title="Doob's martingale inequality">Doob's martingale</a></li> <li><a href="/wiki/Doob%27s_upcrossing_inequality" class="mw-redirect" title="Doob's upcrossing inequality">Doob's upcrossing</a></li> <li><a href="/wiki/Kunita%E2%80%93Watanabe_inequality" title="Kunita–Watanabe inequality">Kunita–Watanabe</a></li> <li><a href="/wiki/Marcinkiewicz%E2%80%93Zygmund_inequality" title="Marcinkiewicz–Zygmund inequality">Marcinkiewicz–Zygmund</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tools</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cameron%E2%80%93Martin_formula" class="mw-redirect" title="Cameron–Martin formula">Cameron–Martin formula</a></li> <li><a href="/wiki/Convergence_of_random_variables" title="Convergence of random variables">Convergence of random variables</a></li> <li><a href="/wiki/Dol%C3%A9ans-Dade_exponential" title="Doléans-Dade exponential">Doléans-Dade exponential</a></li> <li><a href="/wiki/Doob_decomposition_theorem" title="Doob decomposition theorem">Doob decomposition theorem</a></li> <li><a href="/wiki/Doob%E2%80%93Meyer_decomposition_theorem" title="Doob–Meyer decomposition theorem">Doob–Meyer decomposition theorem</a></li> <li><a href="/wiki/Doob%27s_optional_stopping_theorem" class="mw-redirect" title="Doob's optional stopping theorem">Doob's optional stopping theorem</a></li> <li><a href="/wiki/Dynkin%27s_formula" title="Dynkin's formula">Dynkin's formula</a></li> <li><a href="/wiki/Feynman%E2%80%93Kac_formula" title="Feynman–Kac formula">Feynman–Kac formula</a></li> <li><a href="/wiki/Filtration_(probability_theory)" title="Filtration (probability theory)">Filtration</a></li> <li><a href="/wiki/Girsanov_theorem" title="Girsanov theorem">Girsanov theorem</a></li> <li><a href="/wiki/Infinitesimal_generator_(stochastic_processes)" title="Infinitesimal generator (stochastic processes)">Infinitesimal generator</a></li> <li><a href="/wiki/It%C3%B4_integral" class="mw-redirect" title="Itô integral">Itô integral</a></li> <li><a href="/wiki/It%C3%B4%27s_lemma" title="Itô's lemma">Itô's lemma</a></li> <li><a href="/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">Karhunen–Loève theorem</a></li> <li><a href="/wiki/Kolmogorov_continuity_theorem" title="Kolmogorov continuity theorem">Kolmogorov continuity theorem</a></li> <li><a href="/wiki/Kolmogorov_extension_theorem" title="Kolmogorov extension theorem">Kolmogorov extension theorem</a></li> <li><a href="/wiki/L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a></li> <li><a href="/wiki/Malliavin_calculus" title="Malliavin calculus">Malliavin calculus</a></li> <li><a href="/wiki/Martingale_representation_theorem" title="Martingale representation theorem">Martingale representation theorem</a></li> <li><a href="/wiki/Optional_stopping_theorem" title="Optional stopping theorem">Optional stopping theorem</a></li> <li><a href="/wiki/Prokhorov%27s_theorem" title="Prokhorov's theorem">Prokhorov's theorem</a></li> <li><a href="/wiki/Quadratic_variation" title="Quadratic variation">Quadratic variation</a></li> <li><a href="/wiki/Reflection_principle_(Wiener_process)" title="Reflection principle (Wiener process)">Reflection principle</a></li> <li><a href="/wiki/Skorokhod_integral" title="Skorokhod integral">Skorokhod integral</a></li> <li><a href="/wiki/Skorokhod%27s_representation_theorem" title="Skorokhod's representation theorem">Skorokhod's representation theorem</a></li> <li><a href="/wiki/Skorokhod_space" class="mw-redirect" title="Skorokhod space">Skorokhod space</a></li> <li><a href="/wiki/Snell_envelope" title="Snell envelope">Snell envelope</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a> <ul><li><a href="/wiki/Tanaka_equation" title="Tanaka equation">Tanaka</a></li></ul></li> <li><a href="/wiki/Stopping_time" title="Stopping time">Stopping time</a></li> <li><a href="/wiki/Stratonovich_integral" title="Stratonovich integral">Stratonovich integral</a></li> <li><a href="/wiki/Uniform_integrability" title="Uniform integrability">Uniform integrability</a></li> <li><a href="/wiki/Usual_hypotheses" class="mw-redirect" title="Usual hypotheses">Usual hypotheses</a></li> <li><a href="/wiki/Wiener_space" class="mw-redirect" title="Wiener space">Wiener space</a> <ul><li><a href="/wiki/Classical_Wiener_space" title="Classical Wiener space">Classical</a></li> <li><a href="/wiki/Abstract_Wiener_space" title="Abstract Wiener space">Abstract</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Disciplines</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a 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Renewal theory - Wikipedia