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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Max-stabile Prozesse</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><b>Max-stabile Prozesse</b> erweitern die mehrdimensionale <a href="Extremwerttheorie" title="Extremwerttheorie">Extremwerttheorie</a> hin zum unendlichdimensionalen Fall. Ähnlich zum ein- und mehrdimensionalen Fall, tritt ein solcher Prozess als Grenzwert der Maxima von angemessen normalisierten unabhängigen Kopien eines stochastischen Prozesses auf.
</p><div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Sei <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>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T}</annotation>
</semantics>
</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> eine beliebige Indexmenge. Ein <a href="Stochastischer_Prozess" title="Stochastischer Prozess">stochastischer Prozess</a> <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>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</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> heißt max-stabil, falls es Normalisierungskonstanten <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 a_{n}(t)>0,b_{n}(t)\in \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>></mo>
<mn>0</mn>
<mo>,</mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a_{n}(t)>0,b_{n}(t)\in \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/770f09c04ca6e6ef8031550a3946d3c588b733a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.776ex; height:2.843ex;" alt="{\displaystyle a_{n}(t)>0,b_{n}(t)\in \mathbb {R} }" loading="lazy"></span> gibt, sodass für unabhängige Kopien <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_{1},\dotsc ,X_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{1},\dotsc ,X_{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae25602d4b1df2584ef4dd94474b2db32c4e970e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.299ex; height:2.509ex;" alt="{\displaystyle X_{1},\dotsc ,X_{n}}" loading="lazy"></span> des Prozesses <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>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</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> gilt<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>
<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 \left({\frac {\max _{i=1}^{n}X_{i}(t)-b_{n}(t)}{a_{n}(t)}},t\in T\right){\overset {d}{=}}(X(t),t\in T)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<munderover>
<mo movablelimits="true" form="prefix">max</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>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mo>,</mo>
<mi>t</mi>
<mo>∈<!-- ∈ --></mo>
<mi>T</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mo>=</mo>
<mi>d</mi>
</mover>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mi>t</mi>
<mo>∈<!-- ∈ --></mo>
<mi>T</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left({\frac {\max _{i=1}^{n}X_{i}(t)-b_{n}(t)}{a_{n}(t)}},t\in T\right){\overset {d}{=}}(X(t),t\in T)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ebe5afa677f161dbf12c89a2eeadd5cb6a9d0ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.283ex; height:6.676ex;" alt="{\displaystyle \left({\frac {\max _{i=1}^{n}X_{i}(t)-b_{n}(t)}{a_{n}(t)}},t\in T\right){\overset {d}{=}}(X(t),t\in T)}" loading="lazy"></span>.</dd></dl>
<p>Die eindimensionalen Randverteilungen eines max-stabilen Prozesses sind durch eine der drei univariaten Extremwertverteilungen gegeben. Im Falle von <a href="Fr%C3%A9chet-Verteilung" title="Fréchet-Verteilung">Fréchet-verteilten</a> Rändern d. h. <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 {P} (X(t)\leq x)=e^{-1/x}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">P</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>≤<!-- ≤ --></mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>x</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (X(t)\leq x)=e^{-1/x}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4341047452fbd358755549ee71f5af3fde5ffa6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.564ex; height:3.343ex;" alt="{\displaystyle \mathbb {P} (X(t)\leq x)=e^{-1/x}}" loading="lazy"></span> können die Normalisierungskonstanten wie folgt gewählt werden: <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 a_{n}=n,b_{n}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>n</mi>
<mo>,</mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a_{n}=n,b_{n}=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0dca635672371ce476b5dc39f2a0d51937d33bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.452ex; height:2.509ex;" alt="{\displaystyle a_{n}=n,b_{n}=0}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Allgemeines">Allgemeines</h2></div>
<p>Seien <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 Y_{i},i=1,\dotsc ,n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y_{i},i=1,\dotsc ,n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9683d376e9b7b436cf044d496fc6a8ea061a18f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.821ex; height:2.509ex;" alt="{\displaystyle Y_{i},i=1,\dotsc ,n}" loading="lazy"></span> unabhängige Kopien des stochastischen Prozesses <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 Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span>. Gibt es nun Normalisierungskonstanten <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 c_{n}(t)>0,d_{n}(t)\in \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>></mo>
<mn>0</mn>
<mo>,</mo>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c_{n}(t)>0,d_{n}(t)\in \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3503cc9b7fc8ed7c645e440a805b6d62a53f3c45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.764ex; height:2.843ex;" alt="{\displaystyle c_{n}(t)>0,d_{n}(t)\in \mathbb {R} }" loading="lazy"></span>, sodass gilt
<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 \max _{i=1}^{n}{\tfrac {Y_{i}(t)-d_{n}(t)}{c_{n}(t)}}\rightarrow X(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munderover>
<mo movablelimits="true" form="prefix">max</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>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mrow>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<msub>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mstyle>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<mi>X</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \max _{i=1}^{n}{\tfrac {Y_{i}(t)-d_{n}(t)}{c_{n}(t)}}\rightarrow X(t)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4bf1a0dbffc9c082454d001ecdaa992dba868c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.216ex; height:5.009ex;" alt="{\displaystyle \max _{i=1}^{n}{\tfrac {Y_{i}(t)-d_{n}(t)}{c_{n}(t)}}\rightarrow X(t)}" 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 n\to \infty }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\to \infty }</annotation>
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<annotation encoding="application/x-tex">{\displaystyle t\in T}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4fe93f70df3818ecca67c2ca44f087483951856.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.317ex; height:2.176ex;" alt="{\displaystyle t\in T}" loading="lazy"></span> und der 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}">
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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 max-stabiler Prozess. Ein max-stabiler Prozess mit einfachen Fréchet-verteilten Rändern kann mithilfe seiner Spektraldarstellung konstruiert werden.<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>
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<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">Maximilian Zott: <cite style="font-style:italic">Extreme Value Theory in Higher Dimensions - Max-Stable Processes and Multivariate Records</cite>. (<a rel="nofollow" class="external text" href="https://opus.bibliothek.uni-wuerzburg.de/frontdoor/index/index/docId/13661">uni-wuerzburg.de</a> [abgerufen am 7. Oktober 2019]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Max-stabile+Prozesse&rft.au=Maximilian+Zott&rft.btitle=Extreme+Value+Theory+in+Higher+Dimensions+-+Max-Stable+Processes+and+Multivariate+Records&rft.genre=book" style="display:none"> </span></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Laurens de Haan: <i>A Spectral Representation for Max-stable Processes.</i> In: <i>The Annals of Probability.</i> 1984.</span>
</li>
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