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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Sprungprozess</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Ein <b>Sprungprozess</b> ist ein spezieller <a href="Stochastischer_Prozess" title="Stochastischer Prozess">stochastischer Prozess</a> und somit ein Untersuchungsobjekt der <a href="Wahrscheinlichkeitstheorie" title="Wahrscheinlichkeitstheorie">Wahrscheinlichkeitstheorie</a>, einem Teilgebiet der <a href="Mathematik" title="Mathematik">Mathematik</a>. Anschaulich zeichnen sich Sprungprozesse dadurch aus, dass ihr Wert eine gewisse (zufällige) Zeit lang konstant bleibt, um dann einen Sprung zu einem weiteren Wert zu machen, auf dem sie wieder eine Zeit lang verharren. Im einfachsten Fall eines Sprungprozesses mit der Indexmenge und Zustandsmenge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span> bilden die <a href="Pfad_(Stochastik)" title="Pfad (Stochastik)">Pfade</a> eines Sprungprozesses eine <a href="Treppenfunktion_(reelle_Funktion)" title="Treppenfunktion (reelle Funktion)">Treppenfunktion</a>.
</p>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>Gegeben sei ein stochastischer Prozess <span class="mwe-math-element mwe-math-element-inline"><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=(X_{t})_{t\in T}}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
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<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
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</msub>
<msub>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>∈<!-- ∈ --></mo>
<mi>T</mi>
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<annotation encoding="application/x-tex">{\displaystyle X=(X_{t})_{t\in T}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/698b49d3abb3b2dfff77d077a5da132725ec1ff7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.717ex; height:2.843ex;" alt="{\displaystyle X=(X_{t})_{t\in T}}" loading="lazy"></span> mit Indexmenge <span class="mwe-math-element mwe-math-element-inline"><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}">
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<mi>T</mi>
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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span> und Werten in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
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<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span>.
</p><p>Dann heißt <span class="mwe-math-element mwe-math-element-inline"><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> ein <i>Sprungprozess</i>, wenn die <a href="Pfad_(Stochastik)" title="Pfad (Stochastik)">Pfade</a> des Prozesses, also die Abbildungen
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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_{\omega }\colon T\to E}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>W</mi>
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<mi>ω<!-- ω --></mi>
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<mo>:<!-- : --></mo>
<mi>T</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>E</mi>
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<annotation encoding="application/x-tex">{\displaystyle W_{\omega }\colon T\to E}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b3679236f31f60491882fb29db45740bffddf135.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.508ex; height:2.509ex;" alt="{\displaystyle W_{\omega }\colon T\to E}" loading="lazy"></span>,</dd></dl>
<p>definiert durch
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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_{\omega }(t):=X_{t}(\omega )}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>W</mi>
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<mi>ω<!-- ω --></mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
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<mi>ω<!-- ω --></mi>
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<annotation encoding="application/x-tex">{\displaystyle W_{\omega }(t):=X_{t}(\omega )}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d2c739ddc3d7a43811d11551f897d540e48af6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.848ex; height:2.843ex;" alt="{\displaystyle W_{\omega }(t):=X_{t}(\omega )}" loading="lazy"></span></dd></dl>
<p>stückweise konstant sind.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>
<p>Eine große Klasse von Sprungprozessen sind die <a href="Z%C3%A4hlprozess" title="Zählprozess">Zählprozesse</a>, zu denen auch der <a href="Poisson-Prozess" title="Poisson-Prozess">Poisson-Prozess</a> gehört. Anschaulich zählen diese die Anzahl der bis zu einem gewissen Zeitpunkt eingetretenen Ereignisse, ähnlich einem <a href="Geigerz%C3%A4hler" class="mw-redirect" title="Geigerzähler">Geigerzähler</a>. Bei jedem eingetretenen Ereignis springen sie um den Wert eins nach oben.
</p>
<div class="mw-heading mw-heading2"><h2 id="Bemerkung">Bemerkung</h2></div>
<p>Auch bei Sprungprozessen sind degenerierte Fälle möglich und müssen im Zweifel explizit ausgeschlossen werden. Einer dieser Spezialfälle ist eine sogenannte <b>Explosion</b>. Dabei hat der Sprungprozess in endlicher Zeit unendlich viele Sprünge (nach oben).<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><p>Ein möglicher Pfad solch einer Explosion wäre gegeben durch
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(t):=n\;}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>W</mi>
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<mo>:=</mo>
<mi>n</mi>
<mspace width="thickmathspace"></mspace>
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<annotation encoding="application/x-tex">{\displaystyle W(t):=n\;}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b90a8fb890b572c5e8d6bd6b964168e8b0af104.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.869ex; height:2.843ex;" alt="{\displaystyle W(t):=n\;}" loading="lazy"></span> für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;t\in [-{\tfrac {1}{n}};-{\tfrac {1}{n+1}})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="thickmathspace"></mspace>
<mi>t</mi>
<mo>∈<!-- ∈ --></mo>
<mo stretchy="false">[</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mi>n</mi>
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<mo>;</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</mstyle>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \;t\in [-{\tfrac {1}{n}};-{\tfrac {1}{n+1}})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0852ffd896b08dae9270adc19e5e3173fe288c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.272ex; height:3.676ex;" alt="{\displaystyle \;t\in [-{\tfrac {1}{n}};-{\tfrac {1}{n+1}})}" loading="lazy"></span></dd></dl>
<p>mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in [-1,0)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
<mo>∈<!-- ∈ --></mo>
<mo stretchy="false">[</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t\in [-1,0)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55adf2a8f37964382764fcaab8cb40139133d2d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.399ex; height:2.843ex;" alt="{\displaystyle t\in [-1,0)}" loading="lazy"></span>. Solche Explosionen treten beispielsweise bei der Modellierung von Patientenaufnahmen in einem Krankenhaus bei Ausbruch einer Seuche auf. Dabei werden in immer kürzer werdenden Abständen Patienten in das Krankenhaus eingeliefert. Im obigen Beispiel wäre der zeitliche Abstand zwischen Patient <span class="mwe-math-element mwe-math-element-inline"><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-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle n-1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbd0b0f32b28f51962943ee9ede4fb34198a2521.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}" loading="lazy"></span> und Patient <span class="mwe-math-element mwe-math-element-inline"><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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy"></span> genau <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{n}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mi>n</mi>
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</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{n}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee46f3d1f145f31319826905e4ce0750792d55b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.822ex; height:3.343ex;" alt="{\displaystyle {\tfrac {1}{n}}}" loading="lazy"></span> Zeiteinheiten lang. Die Anzahl der belegten Betten zum Zeitpunkt <span class="mwe-math-element mwe-math-element-inline"><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>
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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" 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}" loading="lazy"></span> (vor der Explosion) ist durch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>W</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle W(t)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6bbc0342703bb50b33430da9dcd6c3445e05347.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.084ex; height:2.843ex;" alt="{\displaystyle W(t)}" loading="lazy"></span> gegeben.
</p>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Yu.M. Kabanov: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Jump Process</cite>. In: <a href="Michiel_Hazewinkel" title="Michiel Hazewinkel">Michiel Hazewinkel</a> (Hrsg.): <cite class="lang" lang="en" dir="auto" style="font-style:italic"><a href="Encyclopedia_of_Mathematics" class="mw-redirect" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></cite>. Springer-Verlag und <a href="European_Mathematical_Society" title="European Mathematical Society">EMS</a> Press, Berlin 2002, ISBN 1-55608-010-7 (englisch, <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Jump_process">encyclopediaofmath.org</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rfr_id=info:sid/de.wikipedia.org:Sprungprozess&amp;rft.atitle=Jump+Process&amp;rft.au=Yu.M.+Kabanov&amp;rft.btitle=Encyclopedia+of+Mathematics&amp;rft.date=2002&amp;rft.genre=book&amp;rft.isbn=1556080107&amp;rft.place=Berlin&amp;rft.pub=Springer-Verlag+und+EMS+Press" style="display:none">&nbsp;</span></span>
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<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">David Meintrup, Stefan Schäffler: <cite style="font-style:italic">Stochastik</cite>. Theorie und Anwendungen. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 3-540-21676-6, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em">&nbsp;</span>273–274</span>, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/b137972">10.1007/b137972</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rfr_id=info:sid/de.wikipedia.org:Sprungprozess&amp;rft.au=David+Meintrup%2C+Stefan+Sch%C3%A4ffler&amp;rft.btitle=Stochastik&amp;rft.date=2005&amp;rft.doi=10.1007%2Fb137972&amp;rft.genre=book&amp;rft.isbn=3540216766&amp;rft.pages=273-274&amp;rft.place=Berlin+Heidelberg+New+York&amp;rft.pub=Springer-Verlag" style="display:none">&nbsp;</span></span>
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