Konvergenz (Stochastik)

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Anders als im Fall reeller <a href="/wiki/Zahlenfolge" class="mw-redirect" title="Zahlenfolge">Zahlenfolgen</a> gibt es keine natürliche Definition für das Grenzverhalten von Zufallsvariablen bei wachsendem <a href="/wiki/Stichprobenumfang" class="mw-redirect" title="Stichprobenumfang">Stichprobenumfang</a>, weil das asymptotische Verhalten der Experimente immer von den einzelnen <a href="/wiki/Realisierung_(Stochastik)" title="Realisierung (Stochastik)">Realisierungen</a> abhängt und wir es also formal mit der <a href="/wiki/Funktionenfolge" title="Funktionenfolge">Konvergenz von Funktionen</a> zu tun haben. Daher haben sich im Laufe der Zeit unterschiedlich starke Konzepte herausgebildet, die wichtigsten dieser <b>Konvergenzarten</b> werden im Folgenden kurz vorgestellt. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Voraussetzungen"><span class="tocnumber">1</span> <span class="toctext">Voraussetzungen</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Fast_sichere_Konvergenz"><span class="tocnumber">2</span> <span class="toctext">Fast sichere Konvergenz</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Konvergenz_im_p-ten_Mittel"><span class="tocnumber">3</span> <span class="toctext">Konvergenz im <i>p</i>-ten Mittel</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#Konvergenz_in_Wahrscheinlichkeit"><span class="tocnumber">4</span> <span class="toctext">Konvergenz in Wahrscheinlichkeit</span></a></li> <li class="toclevel-1 tocsection-5"><a href="#Schwache_Konvergenz"><span class="tocnumber">5</span> <span class="toctext">Schwache Konvergenz</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#Zusammenhang_zwischen_den_einzelnen_Konvergenzarten"><span class="tocnumber">6</span> <span class="toctext">Zusammenhang zwischen den einzelnen Konvergenzarten</span></a></li> <li class="toclevel-1 tocsection-7"><a href="#Beispiel"><span class="tocnumber">7</span> <span class="toctext">Beispiel</span></a></li> <li class="toclevel-1 tocsection-8"><a href="#Siehe_auch"><span class="tocnumber">8</span> <span class="toctext">Siehe auch</span></a></li> <li class="toclevel-1 tocsection-9"><a href="#Literatur"><span class="tocnumber">9</span> <span class="toctext">Literatur</span></a></li> <li class="toclevel-1 tocsection-10"><a href="#Einzelnachweise"><span class="tocnumber">10</span> <span class="toctext">Einzelnachweise</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Voraussetzungen">Voraussetzungen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Konvergenz_(Stochastik)&veaction=edit&section=1" title="Abschnitt bearbeiten: Voraussetzungen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Konvergenz_(Stochastik)&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Voraussetzungen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Wir werden die klassischen Konvergenzbegriffe immer im folgenden Modell formulieren: Gegeben sei eine Folge <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_{n})_{(n\in \mathbb {N} )}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </msub> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{n})_{(n\in \mathbb {N} )}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/014978c9d332c7436fec7a8efbfa2dc09f5cff18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.378ex; height:3.176ex;" alt="{\displaystyle (X_{n})_{(n\in \mathbb {N} )}\;}"></span> von Zufallsvariablen, die auf einem <a href="/wiki/Wahrscheinlichkeitsraum" title="Wahrscheinlichkeitsraum">Wahrscheinlichkeitsraum</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,\Sigma ,P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,\Sigma ,P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c3ca06586219fdcc0bb6f1fbb52fe2df0a86d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.979ex; height:2.843ex;" alt="{\displaystyle (\Omega ,\Sigma ,P)}"></span> definiert sind und in denselben <a href="/wiki/Normierter_Raum" title="Normierter Raum">normierten Raum</a> abbilden. Dieser Bildraum wird mit seiner <a href="/wiki/Borelsche_%CF%83-Algebra" title="Borelsche σ-Algebra">Borel-Algebra</a> in natürlicher Weise zu einem <a href="/wiki/Messraum_(Mathematik)" title="Messraum (Mathematik)">Messraum</a>. Um die Kernaussagen zu verstehen, genügt es, sich stets <a href="/wiki/Zufallsvariable#Reelle_Zufallsvariable" title="Zufallsvariable">reelle Zufallsvariablen</a> vorzustellen. Andererseits können die folgenden Definitionen in naheliegender Weise auf den Fall <a href="/wiki/Metrischer_Raum" title="Metrischer Raum">metrischer Räume</a> als Bildraum verallgemeinert werden. </p><p>Eine <a href="/wiki/Stochastischer_Prozess#Pfade" title="Stochastischer Prozess">Realisierung</a> dieser Folge wird üblicherweise mit <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_{n}(\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}(\omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52753fb1ae064177fe35ded2359ab1331a0f76ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.398ex; height:2.843ex;" alt="{\displaystyle X_{n}(\omega )}"></span> bezeichnet. </p> <div class="mw-heading mw-heading2"><h2 id="Fast_sichere_Konvergenz">Fast sichere Konvergenz</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Konvergenz_(Stochastik)&veaction=edit&section=2" title="Abschnitt bearbeiten: Fast sichere Konvergenz" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Konvergenz_(Stochastik)&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Fast sichere Konvergenz"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="/wiki/Fast_sichere_Konvergenz" title="Fast sichere Konvergenz">Fast sichere Konvergenz</a></i></div> <p>Der Begriff der <i>fast sicheren Konvergenz</i> ist am ehesten mit der Formulierung für Zahlenfolgen vergleichbar. Er wird vor allem bei der Formulierung von <a href="/wiki/Starkes_Gesetz_der_gro%C3%9Fen_Zahlen" title="Starkes Gesetz der großen Zahlen">starken Gesetzen der großen Zahlen</a> verwendet. </p><p>Man sagt, dass die Folge <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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8564cedc659cf2f95ae68bc5de2f5207a3285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle X_{n}}"></span> <i>fast sicher</i> gegen eine Zufallsvariable <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}"> <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="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"></span> konvergiert, falls </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 P\left(\lim _{n\to \infty }X_{n}=X\right)=P\left(\left\{\omega \in \Omega \,\left|\,\lim _{n\to \infty }X_{n}(\omega )=X(\omega )\right.\right\}\right)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>X</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mo>{</mo> <mrow> <mi>ω</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mspace width="thinmathspace" /> <mrow> <mo>|</mo> <mrow> <mspace width="thinmathspace" /> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mo>}</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\left(\lim _{n\to \infty }X_{n}=X\right)=P\left(\left\{\omega \in \Omega \,\left|\,\lim _{n\to \infty }X_{n}(\omega )=X(\omega )\right.\right\}\right)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b95557a52511909fb5d1efc5cea0511330be5483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:60.321ex; height:4.843ex;" alt="{\displaystyle P\left(\lim _{n\to \infty }X_{n}=X\right)=P\left(\left\{\omega \in \Omega \,\left|\,\lim _{n\to \infty }X_{n}(\omega )=X(\omega )\right.\right\}\right)=1}"></span></dd></dl> <p>gilt und schreibt dann <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_{n}{\xrightarrow {\text{ f. s. }}}X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mtext> f. s. </mtext> </mpadded> </mover> </mrow> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}{\xrightarrow {\text{ f. s. }}}X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be308b3a4c8c6cbb55ce602e2ad56f62d04ea11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.293ex; width:10.254ex; height:4.009ex;" alt="{\displaystyle X_{n}{\xrightarrow {\text{ f. s. }}}X}"></span>. Übersetzt bedeutet dies, dass für <a href="/wiki/Fast_%C3%BCberall" title="Fast überall">fast alle</a> Realisierungen der Folge der klassische Konvergenzbegriff bezüglich der Norm gilt. Die <i>fast sichere Konvergenz</i> entspricht damit der <i><a href="/wiki/Punktweise_Konvergenz_%CE%BC-fast_%C3%BCberall" title="Punktweise Konvergenz μ-fast überall">punktweisen Konvergenz μ-fast überall</a></i> aus der <a href="/wiki/Ma%C3%9Ftheorie" title="Maßtheorie">Maßtheorie</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Konvergenz_im_p-ten_Mittel">Konvergenz im <i>p</i>-ten Mittel</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Konvergenz_(Stochastik)&veaction=edit&section=3" title="Abschnitt bearbeiten: Konvergenz im p-ten Mittel" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Konvergenz_(Stochastik)&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Konvergenz im p-ten Mittel"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="/wiki/Konvergenz_im_p-ten_Mittel" title="Konvergenz im p-ten Mittel">Konvergenz im p-ten Mittel</a></i></div> <p>Ein <a href="/wiki/Integralrechnung" title="Integralrechnung">integrationstheoretischer</a> Ansatz wird mit dem Begriff der <i>Konvergenz im <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>-ten Mittel</i> verfolgt. Es werden dabei nicht einzelne Realisierungen betrachtet, sondern <a href="/wiki/Erwartungswert" title="Erwartungswert">Erwartungswerte</a> der Zufallsvariablen. </p><p>Formal konvergiert <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_{n}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1089e6920092950d1b661bc07ff62626b169b4dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.788ex; height:2.509ex;" alt="{\displaystyle X_{n}\;}"></span> im <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>-ten Mittel gegen eine Zufallsvariable <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}"> <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="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"></span>, falls </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 _{n\rightarrow \infty }E[|X_{n}-X|^{p}]=0}"> <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>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>E</mi> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>X</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\rightarrow \infty }E[|X_{n}-X|^{p}]=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8989497c5160d9d559c8f80e1ad5ce91ae62626" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.306ex; height:4.009ex;" alt="{\displaystyle \lim _{n\rightarrow \infty }E[|X_{n}-X|^{p}]=0}"></span></dd></dl> <p>gilt. Dabei wird <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\geq 1\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>≥</mo> <mn>1</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\geq 1\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a112be990df9e7cf0dd47f30e414f597cc2709ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.165ex; height:2.509ex;" alt="{\displaystyle p\geq 1\;}"></span> vorausgesetzt. Dies bedeutet, dass die Differenz <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_{n}-X\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>X</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}-X\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0267c80abc8401e5a35d3aa46560d2f76717173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.608ex; height:2.509ex;" alt="{\displaystyle X_{n}-X\;}"></span> im <a href="/wiki/Lp-Raum" title="Lp-Raum"><i>L<sup>p</sup></i>-Raum</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{p}(P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{p}(P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f5b05631122deff04c17c07d69dc94341f3a7a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.218ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}^{p}(P)}"></span> gegen <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" 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 0}"></span> konvergiert. Man bezeichnet diese Konvergenz daher auch als <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74656e06707b0d5ba1cd9fbe51155b561d082e23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.663ex; height:2.343ex;" alt="{\displaystyle {\mathcal {L}}^{p}}"></span>-Konvergenz. </p><p>Wegen der <a href="/wiki/Ungleichung_vom_arithmetischen_und_geometrischen_Mittel" title="Ungleichung vom arithmetischen und geometrischen Mittel">Ungleichung der verallgemeinerten Mittelwerte</a> folgt für <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 q>p\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>></mo> <mi>p</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q>p\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4538adb30d8652de7fde7af7e8a72acfd02fe43f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.983ex; height:2.176ex;" alt="{\displaystyle q>p\;}"></span> aus der Konvergenz im <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 q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>-ten Mittel die Konvergenz im <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>-ten Mittel. </p> <div class="mw-heading mw-heading2"><h2 id="Konvergenz_in_Wahrscheinlichkeit">Konvergenz in Wahrscheinlichkeit</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Konvergenz_(Stochastik)&veaction=edit&section=4" title="Abschnitt bearbeiten: Konvergenz in Wahrscheinlichkeit" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Konvergenz_(Stochastik)&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Konvergenz in Wahrscheinlichkeit"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="/wiki/Konvergenz_in_Wahrscheinlichkeit" title="Konvergenz in Wahrscheinlichkeit">Konvergenz in Wahrscheinlichkeit</a></i></div> <p>Ein etwas schwächerer Konvergenzbegriff ist die <i>stochastische Konvergenz</i> oder <i>Konvergenz in Wahrscheinlichkeit</i>. Wie der Name bereits suggeriert, werden nicht spezielle Realisierungen der Zufallsvariablen betrachtet, sondern <a href="/wiki/Wahrscheinlichkeit" title="Wahrscheinlichkeit">Wahrscheinlichkeiten</a> für bestimmte Ereignisse. Eine klassische Anwendung der stochastischen Konvergenz sind <a href="/wiki/Schwaches_Gesetz_der_gro%C3%9Fen_Zahlen" title="Schwaches Gesetz der großen Zahlen">schwache Gesetze der großen Zahlen</a>. </p><p>Die mathematische Formulierung für <a href="/wiki/Zufallsvariable#Reelle_Zufallsvariable" title="Zufallsvariable">reelle Zufallsvariablen</a> lautet: Eine Folge <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_{n})_{n\in \mathbb {N} }\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{n})_{n\in \mathbb {N} }\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961fcff08ff58ea2e534f17c386a1ca78aa07116" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.098ex; height:2.843ex;" alt="{\displaystyle (X_{n})_{n\in \mathbb {N} }\;}"></span> von reellen Zufallsvariablen konvergiert stochastisch gegen eine Zufallsvariable <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}"> <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="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"></span>, falls </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 \forall \varepsilon >0\colon \lim _{n\to \infty }P(\vert X_{n}-X\vert >\varepsilon )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mo>:</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>P</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">|</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>X</mi> <mo fence="false" stretchy="false">|</mo> <mo>></mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \varepsilon >0\colon \lim _{n\to \infty }P(\vert X_{n}-X\vert >\varepsilon )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc66c332965fc430067a9dafbd2431b02350d3d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.585ex; height:3.843ex;" alt="{\displaystyle \forall \varepsilon >0\colon \lim _{n\to \infty }P(\vert X_{n}-X\vert >\varepsilon )=0}"></span></dd></dl> <p>gilt. Für die Konvergenz in Wahrscheinlichkeit werden meist folgende Schreibweisen verwendet: <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-lim} _{n\rightarrow \infty }\,X_{n}=X\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">P</mi> <mtext>-</mtext> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>X</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {P-lim} _{n\rightarrow \infty }\,X_{n}=X\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/836ba10951f6e35a2c255c7a2fd0fad44e030959" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.345ex; height:2.509ex;" alt="{\displaystyle \operatorname {P-lim} _{n\rightarrow \infty }\,X_{n}=X\;}"></span> oder <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 {plim} (X_{n})=X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>plim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {plim} (X_{n})=X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/777ee8e390ef36416847c069cf803911c34ba2b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.553ex; height:2.843ex;" alt="{\displaystyle \operatorname {plim} (X_{n})=X}"></span> oder <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_{n}{\stackrel {P}{\rightarrow }}X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </mover> </mrow> </mrow> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}{\stackrel {P}{\rightarrow }}X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32717055ec7538300a75ff95bc7e43a812b7a32a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.446ex; height:4.009ex;" alt="{\displaystyle X_{n}{\stackrel {P}{\rightarrow }}X}"></span>. </p><p>Die <i>stochastische Konvergenz</i> entspricht der <i><a href="/wiki/Konvergenz_dem_Ma%C3%9Fe_nach" class="mw-redirect" title="Konvergenz dem Maße nach">Konvergenz dem Maße nach</a></i> aus der Maßtheorie. </p> <div class="mw-heading mw-heading2"><h2 id="Schwache_Konvergenz">Schwache Konvergenz</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Konvergenz_(Stochastik)&veaction=edit&section=5" title="Abschnitt bearbeiten: Schwache Konvergenz" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Konvergenz_(Stochastik)&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Schwache Konvergenz"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="/wiki/Konvergenz_in_Verteilung" title="Konvergenz in Verteilung">Konvergenz in Verteilung</a></i></div> <p>Der vierte prominente Konvergenzbegriff ist der der <i>Konvergenz in Verteilung</i>, manchmal auch schwache Konvergenz (für Zufallsvariablen) genannt. Er entspricht der <i><a href="/wiki/Schwache_Konvergenz_(Ma%C3%9Ftheorie)" title="Schwache Konvergenz (Maßtheorie)">schwachen Konvergenz für Maße</a></i> der Maßtheorie. </p><p>Eine Folge von Zufallsvariablen <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_{n}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1089e6920092950d1b661bc07ff62626b169b4dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.788ex; height:2.509ex;" alt="{\displaystyle X_{n}\;}"></span> konvergiert in Verteilung gegen die Zufallsvariable <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}"> <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="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"></span>, wenn die Folge der induzierten Bildmaße <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 _{n}(A):=P(X_{n}\in A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{n}(A):=P(X_{n}\in A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14f04381e14208daa7ba23a1ebb7757d3176ca9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.199ex; height:2.843ex;" alt="{\displaystyle \mu _{n}(A):=P(X_{n}\in A)}"></span> schwach gegen das Bildmaß <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 (A):=P(X\in A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>∈</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (A):=P(X\in A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d13caed8838b3c70a3af512fb18609d8c0f2af7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.818ex; height:2.843ex;" alt="{\displaystyle \mu (A):=P(X\in A)}"></span> konvergiert. Das heißt, für alle stetigen beschränkten Funktionen <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> gilt </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 _{n\to \infty }E(f\circ X_{n})=E(f\circ X)}"> <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>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∘</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }E(f\circ X_{n})=E(f\circ X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/958fae8008d50b58034ae14715cb880fb281397a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.997ex; height:3.843ex;" alt="{\displaystyle \lim _{n\to \infty }E(f\circ X_{n})=E(f\circ X)}"></span>.</dd></dl> <p>Für reelle Zufallsvariable ist nach dem <a href="/wiki/Satz_von_Helly-Bray" title="Satz von Helly-Bray">Satz von Helly-Bray</a> die folgende Charakterisierung äquivalent dazu: Für die <a href="/wiki/Verteilungsfunktion" title="Verteilungsfunktion">Verteilungsfunktionen</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"></span> von <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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8564cedc659cf2f95ae68bc5de2f5207a3285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle X_{n}}"></span> und <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> von <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}"> <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="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"></span> gilt </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 _{n\to \infty }F_{n}(x)=F(x)}"> <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>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }F_{n}(x)=F(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/732e5a10862bfd7c3d470e681f53f4cd11070a15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.49ex; height:3.843ex;" alt="{\displaystyle \lim _{n\to \infty }F_{n}(x)=F(x)}"></span></dd></dl> <p>an allen Stellen <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\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c6d458566aec47a7259762034790c8981aefab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {R} }"></span>, an denen <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> stetig ist.<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> Die wohl bekanntesten Anwendungen der Konvergenz in Verteilung sind <a href="/wiki/Zentraler_Grenzwertsatz" title="Zentraler Grenzwertsatz">zentrale Grenzwertsätze</a>. </p><p>Da die Konvergenz in Verteilung ausschließlich durch die Bildmaße bzw. durch die Verteilungsfunktion der Zufallsvariablen definiert sind, ist es nicht notwendig, dass die Zufallsvariablen auf demselben Wahrscheinlichkeitsraum definiert sind. </p><p>Als Notation verwendet man in der Regel <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_{n}{\stackrel {w}{\rightarrow }}X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </mover> </mrow> </mrow> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}{\stackrel {w}{\rightarrow }}X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44d170c56e6d0a18c923386bbd2c94c76f813da2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.446ex; height:3.509ex;" alt="{\displaystyle X_{n}{\stackrel {w}{\rightarrow }}X}"></span> oder <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_{n}{\stackrel {\mathcal {D}}{\rightarrow }}X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}{\stackrel {\mathcal {D}}{\rightarrow }}X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e927c2b61902d9bbca38d7d8159fcb1b8aad42e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.446ex; height:4.009ex;" alt="{\displaystyle X_{n}{\stackrel {\mathcal {D}}{\rightarrow }}X}"></span>, manchmal aber auch <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_{n}\implies X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}\implies X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/865d4ae6a15524b3effcea6e9e3a4d55c36de394" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.509ex; height:2.509ex;" alt="{\displaystyle X_{n}\implies X}"></span>. Die Buchstaben „W“ bzw. „D“ stehen dabei für die entsprechenden Begriffe im Englischen, also <i>weak convergence</i> bzw. <i>convergence in distribution</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Zusammenhang_zwischen_den_einzelnen_Konvergenzarten">Zusammenhang zwischen den einzelnen Konvergenzarten</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Konvergenz_(Stochastik)&veaction=edit&section=6" title="Abschnitt bearbeiten: Zusammenhang zwischen den einzelnen Konvergenzarten" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Konvergenz_(Stochastik)&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Zusammenhang zwischen den einzelnen Konvergenzarten"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In der Reihe der wichtigsten Konvergenzbegriffe in der Stochastik stellen die beiden zuerst vorgestellten Begriffe die stärksten Konvergenzarten dar. Sowohl aus <i>fast sicherer Konvergenz</i><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> als auch aus <i>Konvergenz im p-ten Mittel</i><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> lässt sich immer die <i>stochastische Konvergenz</i> einer Folge von Zufallsvariablen ableiten. Ferner folgt aus <i>stochastischer Konvergenz</i> automatisch auch die <i>Konvergenz in Verteilung</i>, die die schwächste der hier vorgestellten Konvergenzarten ist.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Kompakt gilt also </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{matrix}{\text{Fast sichere}}\\{\text{Konvergenz}}\end{matrix}}\implies {\begin{matrix}{\text{Konvergenz in}}\\{\text{Wahrscheinlichkeit}}\end{matrix}}\implies {\begin{matrix}{\text{Konvergenz in}}\\{\text{Verteilung}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Fast sichere</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Konvergenz</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Konvergenz in</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Wahrscheinlichkeit</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Konvergenz in</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Verteilung</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\text{Fast sichere}}\\{\text{Konvergenz}}\end{matrix}}\implies {\begin{matrix}{\text{Konvergenz in}}\\{\text{Wahrscheinlichkeit}}\end{matrix}}\implies {\begin{matrix}{\text{Konvergenz in}}\\{\text{Verteilung}}\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82a82335653f31c8931ba4796edc1a820bae5e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:60.854ex; height:6.176ex;" alt="{\displaystyle {\begin{matrix}{\text{Fast sichere}}\\{\text{Konvergenz}}\end{matrix}}\implies {\begin{matrix}{\text{Konvergenz in}}\\{\text{Wahrscheinlichkeit}}\end{matrix}}\implies {\begin{matrix}{\text{Konvergenz in}}\\{\text{Verteilung}}\end{matrix}}}"></span></dd></dl> <p>und </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{matrix}{\text{Konvergenz im}}\\{\text{p-ten Mittel}}\end{matrix}}\implies {\begin{matrix}{\text{Konvergenz in}}\\{\text{Wahrscheinlichkeit}}\end{matrix}}\implies {\begin{matrix}{\text{Konvergenz in}}\\{\text{Verteilung}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Konvergenz im</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>p-ten Mittel</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Konvergenz in</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Wahrscheinlichkeit</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Konvergenz in</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Verteilung</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\text{Konvergenz im}}\\{\text{p-ten Mittel}}\end{matrix}}\implies {\begin{matrix}{\text{Konvergenz in}}\\{\text{Wahrscheinlichkeit}}\end{matrix}}\implies {\begin{matrix}{\text{Konvergenz in}}\\{\text{Verteilung}}\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bd940702496ee9a01638568b02eff756a617394" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:64.017ex; height:6.176ex;" alt="{\displaystyle {\begin{matrix}{\text{Konvergenz im}}\\{\text{p-ten Mittel}}\end{matrix}}\implies {\begin{matrix}{\text{Konvergenz in}}\\{\text{Wahrscheinlichkeit}}\end{matrix}}\implies {\begin{matrix}{\text{Konvergenz in}}\\{\text{Verteilung}}\end{matrix}}}"></span>.</dd></dl> <p>In Ausnahmefällen gelten auch noch andere Implikationen: Wenn eine Folge von Zufallsvariablen in Verteilung gegen eine Zufallsvariable X konvergiert und X fast sicher konstant ist, dann konvergiert diese Folge auch stochastisch. </p><p>Aus der Konvergenz im p-ten Mittel folgt im Allgemeinen nicht die fast sichere Konvergenz. Umgekehrt lässt sich aus fast sicherer Konvergenz im Allgemeinen auch keine Konvergenz im p-ten Mittel schließen. Allerdings ist dieser Schluss erlaubt, wenn es eine gemeinsame Majorante in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2317aaca1ecee4b8ccf667bc1001059eae5850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.642ex; height:2.343ex;" alt="{\displaystyle L^{p}}"></span> gibt (siehe <a href="/wiki/Satz_von_der_majorisierten_Konvergenz" title="Satz von der majorisierten Konvergenz">Satz von der majorisierten Konvergenz</a>). Eine Folge von Zufallsvariablen konvergiert genau dann in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74c288d1089f1ec85b01b4de25c441fc792bd2d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{1}}"></span>, wenn sie stochastisch konvergiert und <a href="/wiki/Gleichgradige_Integrierbarkeit" title="Gleichgradige Integrierbarkeit">gleichgradig integrierbar</a> ist. </p> <div class="mw-heading mw-heading2"><h2 id="Beispiel">Beispiel</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Konvergenz_(Stochastik)&veaction=edit&section=7" title="Abschnitt bearbeiten: Beispiel" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Konvergenz_(Stochastik)&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Beispiel"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Auf dem Wahrscheinlichkeitsraum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,\Sigma ,P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,\Sigma ,P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c3ca06586219fdcc0bb6f1fbb52fe2df0a86d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.979ex; height:2.843ex;" alt="{\displaystyle (\Omega ,\Sigma ,P)}"></span> mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega =[0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =[0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a16b39940ed262a459907035d290bf78c18134b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.429ex; height:2.843ex;" alt="{\displaystyle \Omega =[0,1]}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" 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 \Sigma }"></span> den <a href="/wiki/Borelsche_%CF%83-Algebra" title="Borelsche σ-Algebra">Borelmengen</a> und <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/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> dem <a href="/wiki/Borel-Lebesgue-Ma%C3%9F" class="mw-redirect" title="Borel-Lebesgue-Maß">Borel-Lebesgue-Maß</a> betrachte man die Zufallsvariable <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(\omega )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(\omega )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1f5905512524bc3b10f3571f49dc9684f5b292b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.496ex; height:2.843ex;" alt="{\displaystyle X(\omega )=0}"></span> sowie die Folge <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_{n}(\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}(\omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52753fb1ae064177fe35ded2359ab1331a0f76ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.398ex; height:2.843ex;" alt="{\displaystyle X_{n}(\omega )}"></span> der Zufallsvariablen, die für <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=2^{k}+m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>+</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2^{k}+m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca80f8601a9a36a3881d839a885cf45d4bdda536" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.625ex; height:2.843ex;" alt="{\displaystyle n=2^{k}+m}"></span> mit <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\leq m<2^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>m</mi> <mo><</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq m<2^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c78f13c64cc3565353975140dd1b271484be2849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.651ex; height:2.843ex;" alt="{\displaystyle 0\leq m<2^{k}}"></span> (jedes natürliche <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> besitzt eine eindeutige Zerlegung dieser Art) folgendermaßen definiert ist<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-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 X_{n}(\omega )={\begin{cases}1&{\text{falls }}{\frac {m}{2^{k}}}\leq \omega \leq {\frac {m+1}{2^{k}}}\\0&{\text{sonst.}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <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> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>falls </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mfrac> </mrow> <mo>≤</mo> <mi>ω</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>sonst.</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}(\omega )={\begin{cases}1&{\text{falls }}{\frac {m}{2^{k}}}\leq \omega \leq {\frac {m+1}{2^{k}}}\\0&{\text{sonst.}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fd5d8026bdb819a17505789b4c5c61cc8fcd95c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.811ex; height:7.509ex;" alt="{\displaystyle X_{n}(\omega )={\begin{cases}1&{\text{falls }}{\frac {m}{2^{k}}}\leq \omega \leq {\frac {m+1}{2^{k}}}\\0&{\text{sonst.}}\end{cases}}}"></span></dd></dl> <p>Die Funktionen <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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8564cedc659cf2f95ae68bc5de2f5207a3285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle X_{n}}"></span> sind sozusagen immer dünner werdende Zacken, die über das Intervall <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 [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}"></span> laufen. </p><p>Wegen </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 E[|X_{n}-X|^{p}]=\int _{0}^{1}|X_{n}(\omega )-0|^{p}d\omega ={\frac {1}{2^{k}}}\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>X</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>0</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mi>d</mi> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mfrac> </mrow> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[|X_{n}-X|^{p}]=\int _{0}^{1}|X_{n}(\omega )-0|^{p}d\omega ={\frac {1}{2^{k}}}\to 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eff61ad651fb2cfc80095a0137bd87a5b114b655" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:46.767ex; height:6.176ex;" alt="{\displaystyle E[|X_{n}-X|^{p}]=\int _{0}^{1}|X_{n}(\omega )-0|^{p}d\omega ={\frac {1}{2^{k}}}\to 0}"></span></dd></dl> <p>konvergiert <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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8564cedc659cf2f95ae68bc5de2f5207a3285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle X_{n}}"></span> im p-ten Mittel gegen <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}"> <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="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"></span>. Aus dem oben beschriebenen Zusammenhang zwischen den einzelnen Konvergenzarten folgt, dass <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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8564cedc659cf2f95ae68bc5de2f5207a3285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle X_{n}}"></span> ebenso stochastisch gegen <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}"> <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="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"></span> konvergiert, wie sich auch aus </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 P(|X_{n}-X|>\varepsilon )={\begin{cases}{\frac {1}{2^{k}}}\;&{\text{für}}\;0<\varepsilon \leq 1\\0\;&{\text{für}}\;\varepsilon >1\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mfrac> </mrow> <mspace width="thickmathspace" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>für</mtext> </mrow> <mspace width="thickmathspace" /> <mn>0</mn> <mo><</mo> <mi>ε</mi> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mspace width="thickmathspace" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>für</mtext> </mrow> <mspace width="thickmathspace" /> <mi>ε</mi> <mo>></mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(|X_{n}-X|>\varepsilon )={\begin{cases}{\frac {1}{2^{k}}}\;&{\text{für}}\;0<\varepsilon \leq 1\\0\;&{\text{für}}\;\varepsilon >1\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdcd4922560e6066a762d7f633193d2c2bb9e53e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.493ex; height:7.509ex;" alt="{\displaystyle P(|X_{n}-X|>\varepsilon )={\begin{cases}{\frac {1}{2^{k}}}\;&{\text{für}}\;0<\varepsilon \leq 1\\0\;&{\text{für}}\;\varepsilon >1\end{cases}}}"></span></dd></dl> <p>und wegen <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 k\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/612a3ec99f1c9f12de1cfab011e306ae799858ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.149ex; height:2.176ex;" alt="{\displaystyle k\rightarrow \infty }"></span> für <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\rightarrow \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\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9702f04f2d0e5b887b99faeeffb0c4cfd8263eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\rightarrow \infty }"></span>, also </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 P(|X_{n}-X|>\varepsilon )\leq {\frac {1}{2^{k}}}\to 0\;{\text{für jedes}}\;\varepsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mfrac> </mrow> <mo stretchy="false">→</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>für jedes</mtext> </mrow> <mspace width="thickmathspace" /> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(|X_{n}-X|>\varepsilon )\leq {\frac {1}{2^{k}}}\to 0\;{\text{für jedes}}\;\varepsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f047ed620f5c0f3d156df6cb405bc4b7d03e9938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.197ex; height:5.676ex;" alt="{\displaystyle P(|X_{n}-X|>\varepsilon )\leq {\frac {1}{2^{k}}}\to 0\;{\text{für jedes}}\;\varepsilon >0}"></span></dd></dl> <p>erkennen lässt. </p><p>Für jedes fixe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \in [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega \in [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32b70a2d779c8b8d1ea321e7eb870979b88b0e26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.939ex; height:2.843ex;" alt="{\displaystyle \omega \in [0,1]}"></span> gilt aber <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_{n}(\omega )=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}(\omega )=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52c0981dfa9bba8af9236221cf61e6db3e2d8430" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.659ex; height:2.843ex;" alt="{\displaystyle X_{n}(\omega )=1}"></span> für unendliche viele <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>, ebenso ist <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_{n}(\omega )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}(\omega )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecc3a7929c9bb9350f660ccae64fd938bb1d61ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.659ex; height:2.843ex;" alt="{\displaystyle X_{n}(\omega )=0}"></span> für unendlich viele <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>, sodass also keine fast sichere Konvergenz von <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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8564cedc659cf2f95ae68bc5de2f5207a3285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle X_{n}}"></span> vorliegt. Zu jeder Teilfolge <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_{n_{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n_{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/265e80874e182e6c2acbe8b5435083026c23286b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.768ex; height:2.843ex;" alt="{\displaystyle X_{n_{i}}}"></span> von <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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8564cedc659cf2f95ae68bc5de2f5207a3285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle X_{n}}"></span> lässt sich allerdings eine Teilteilfolge <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_{n_{i_{j}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n_{i_{j}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdcbc30ca4135d9d6a2f919388e988e437d00eb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:4.451ex; height:3.509ex;" alt="{\displaystyle X_{n_{i_{j}}}}"></span> finden, die fast sicher gegen <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}"> <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="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"></span> konvergiert. Gäbe es eine <a href="/wiki/Topologie_(Mathematik)" title="Topologie (Mathematik)">Topologie</a> der fast sicheren Konvergenz, so würde aus dieser Eigenschaft folgen, dass <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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a8564cedc659cf2f95ae68bc5de2f5207a3285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle X_{n}}"></span> fast sicher gegen <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}"> <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="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" 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}"></span> konvergiert. Dieses Beispiel zeigt also auch, dass es keine Topologie der fast sicheren Konvergenz geben kann.<sup id="cite_ref-Cigler/Reichel_7-0" class="reference"><a href="#cite_note-Cigler/Reichel-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Konvergenz_(Stochastik)&veaction=edit&section=8" title="Abschnitt bearbeiten: Siehe auch" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Konvergenz_(Stochastik)&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Siehe auch"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Funktionenfolge#Maßtheoretische_Konvergenzbegriffe" title="Funktionenfolge">Maßtheoretische Konvergenzbegriffe</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Konvergenz_(Stochastik)&veaction=edit&section=9" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Konvergenz_(Stochastik)&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Heinz_Bauer_(Mathematiker)" title="Heinz Bauer (Mathematiker)">Heinz Bauer</a>: <cite style="font-style:italic">Wahrscheinlichkeitstheorie</cite>. 4. Auflage. De Gruyter, Berlin 1991, <a href="/wiki/Spezial:ISBN-Suche/3110121905" class="internal mw-magiclink-isbn">ISBN 3-11-012190-5</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>34</span> (Konvergenz von Zufallsvariablen und Verteilungen).<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:Konvergenz+%28Stochastik%29&rft.au=Heinz+Bauer&rft.btitle=Wahrscheinlichkeitstheorie&rft.date=1991&rft.edition=4.&rft.genre=book&rft.isbn=3110121905&rft.pages=34&rft.place=Berlin&rft.pub=De+Gruyter" style="display:none"> </span></li> <li>Heinz Bauer: <cite style="font-style:italic">Maß- und Integrationstheorie</cite>. 2. Auflage. De Gruyter, Berlin 1992, <a href="/wiki/Spezial:ISBN-Suche/3110136252" class="internal mw-magiclink-isbn">ISBN 3-11-013625-2</a>, §15 Konvergenzsätze und §20 Stochastische Konvergenz, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>91<span style="display:inline-block;width:.2em"> </span>ff. und 128<span style="display:inline-block;width:.2em"> </span>ff</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Konvergenz+%28Stochastik%29&rft.atitle=%C2%A715+Konvergenzs%C3%A4tze+und+%C2%A720+Stochastische+Konvergenz&rft.au=Heinz+Bauer&rft.btitle=Ma%C3%9F-+und+Integrationstheorie&rft.date=1992&rft.edition=2.&rft.genre=bookitem&rft.isbn=3110136252&rft.pages=91+ff.+und+128+ff.&rft.place=Berlin&rft.pub=De+Gruyter" style="display:none"> </span></li> <li><a href="/wiki/J%C3%BCrgen_Elstrodt" title="Jürgen Elstrodt">Jürgen Elstrodt</a>: <cite style="font-style:italic">Maß- und Integrationstheorie</cite>. 7. Auflage. Springer, Berlin 2011, <a href="/wiki/Spezial:ISBN-Suche/9783642179044" class="internal mw-magiclink-isbn">ISBN 978-3-642-17904-4</a>, Kapitel VI. Konvergenzbegriffe der Maß- und Integrationstheorie, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>219–268</span> (beschreibt ausführlich die Zusammenhänge zwischen den verschiedenen Konvergenzarten).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Konvergenz+%28Stochastik%29&rft.atitle=Kapitel+VI.+Konvergenzbegriffe+der+Ma%C3%9F-+und+Integrationstheorie&rft.au=J%C3%BCrgen+Elstrodt&rft.btitle=Ma%C3%9F-+und+Integrationstheorie&rft.date=2011&rft.edition=7.&rft.genre=bookitem&rft.isbn=9783642179044&rft.pages=219-268&rft.place=Berlin&rft.pub=Springer" style="display:none"> </span></li> <li><a href="/wiki/Christian_Hesse_(Mathematiker)" title="Christian Hesse (Mathematiker)">Christian Hesse</a>: <cite style="font-style:italic">Angewandte Wahrscheinlichkeitstheorie</cite>. 1. Auflage. Vieweg, Wiesbaden 2003, <a href="/wiki/Spezial:ISBN-Suche/3528031832" class="internal mw-magiclink-isbn">ISBN 3-528-03183-2</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>216–238</span>, <a href="/wiki/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/10.1007/978-3-663-01244-3">10.1007/978-3-663-01244-3</a></span>.<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:Konvergenz+%28Stochastik%29&rft.au=Christian+Hesse&rft.btitle=Angewandte+Wahrscheinlichkeitstheorie&rft.date=2003&rft.doi=10.1007%2F978-3-663-01244-3&rft.edition=1.&rft.genre=book&rft.isbn=3528031832&rft.pages=216-238&rft.place=Wiesbaden&rft.pub=Vieweg" style="display:none"> </span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Konvergenz_(Stochastik)&veaction=edit&section=10" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Konvergenz_(Stochastik)&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></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">Robert B. Ash: <cite style="font-style:italic">Real Analysis and Probability</cite>. Academic Press, New York 1972, <a href="/wiki/Spezial:ISBN-Suche/0120652013" class="internal mw-magiclink-isbn">ISBN 0-12-065201-3</a>, Theorem 4.5.4.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Konvergenz+%28Stochastik%29&rft.atitle=Theorem+4.5.4&rft.au=Robert+B.+Ash&rft.btitle=Real+Analysis+and+Probability&rft.date=1972&rft.genre=bookitem&rft.isbn=0120652013&rft.place=New+York&rft.pub=Academic+Press" 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">Robert B. Ash: <cite style="font-style:italic">Real Analysis and Probability</cite>. Academic Press, New York 1972, <a href="/wiki/Spezial:ISBN-Suche/0120652013" class="internal mw-magiclink-isbn">ISBN 0-12-065201-3</a>, Theorem 2.5.5.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Konvergenz+%28Stochastik%29&rft.atitle=Theorem+2.5.5&rft.au=Robert+B.+Ash&rft.btitle=Real+Analysis+and+Probability&rft.date=1972&rft.genre=bookitem&rft.isbn=0120652013&rft.place=New+York&rft.pub=Academic+Press" style="display:none"> </span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Robert B. Ash: <cite style="font-style:italic">Real Analysis and Probability</cite>. Academic Press, New York 1972, <a href="/wiki/Spezial:ISBN-Suche/0120652013" class="internal mw-magiclink-isbn">ISBN 0-12-065201-3</a>, Theorem 2.5.1.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Konvergenz+%28Stochastik%29&rft.atitle=Theorem+2.5.1&rft.au=Robert+B.+Ash&rft.btitle=Real+Analysis+and+Probability&rft.date=1972&rft.genre=bookitem&rft.isbn=0120652013&rft.place=New+York&rft.pub=Academic+Press" style="display:none"> </span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.math.uah.edu/stat/dist/Convergence.html">Virtual Laboratories in Probability and Statistics, Excercise 2.8.3</a></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Robert B. Ash: <cite style="font-style:italic">Real Analysis and Probability</cite>. Academic Press, New York 1972, <a href="/wiki/Spezial:ISBN-Suche/0120652013" class="internal mw-magiclink-isbn">ISBN 0-12-065201-3</a>, Examples 2.5.6.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Konvergenz+%28Stochastik%29&rft.atitle=Examples+2.5.6&rft.au=Robert+B.+Ash&rft.btitle=Real+Analysis+and+Probability&rft.date=1972&rft.genre=bookitem&rft.isbn=0120652013&rft.place=New+York&rft.pub=Academic+Press" style="display:none"> </span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">Bernard R. Gelbaum, John M.H. Olmsted: <cite style="font-style:italic">Counterexamples in Analysis</cite>. Dover Publications, Mineola, New York 2003, <a href="/wiki/Spezial:ISBN-Suche/0486428753" class="internal mw-magiclink-isbn">ISBN 0-486-42875-3</a>, Abschnitt 8.40, Sequences of functions converging in different senses, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>109–111</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Konvergenz+%28Stochastik%29&rft.atitle=Abschnitt+8.40%2C+Sequences+of+functions+converging+in+different+senses&rft.au=Bernard+R.+Gelbaum%2C+John+M.H.+Olmsted&rft.btitle=Counterexamples+in+Analysis&rft.date=2003&rft.genre=bookitem&rft.isbn=0486428753&rft.pages=109-111&rft.place=Mineola%2C+New+York&rft.pub=Dover+Publications" style="display:none"> </span></span> </li> <li id="cite_note-Cigler/Reichel-7"><span class="mw-cite-backlink"><a href="#cite_ref-Cigler/Reichel_7-0">↑</a></span> <span class="reference-text">J. Cigler, H.-C. Reichel: <cite style="font-style:italic">Topologie. Eine Grundvorlesung</cite>. 6. Auflage. Bibliographisches Institut, Mannheim 1978, <a href="/wiki/Spezial:ISBN-Suche/3411001216" class="internal mw-magiclink-isbn">ISBN 3-411-00121-6</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>88</span>.<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:Konvergenz+%28Stochastik%29&rft.au=J.+Cigler%2C+H.-C.+Reichel&rft.btitle=Topologie.+Eine+Grundvorlesung&rft.date=1978&rft.edition=6&rft.genre=book&rft.isbn=3411001216&rft.pages=88&rft.place=Mannheim&rft.pub=Bibliographisches+Institut" style="display:none"> </span></span> </li> </ol></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Konvergenz_(Stochastik)&oldid=237729528">https://de.wikipedia.org/w/index.php?title=Konvergenz_(Stochastik)&oldid=237729528</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorien</a>: <ul><li><a href="/wiki/Kategorie:Stochastik" title="Kategorie:Stochastik">Stochastik</a></li><li><a href="/wiki/Kategorie:Konvergenzbegriff" title="Kategorie:Konvergenzbegriff">Konvergenzbegriff</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Meine Werkzeuge</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="Benutzerseite der IP-Adresse, von der aus du Änderungen durchführst">Nicht angemeldet</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n"><span>Diskussionsseite</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y"><span>Beiträge</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=Konvergenz+%28Stochastik%29" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. 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class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Converg%C3%A8ncia_de_variables_aleat%C3%B2ries" title="Convergència de variables aleatòries – Katalanisch" lang="ca" hreflang="ca" data-title="Convergència de variables aleatòries" data-language-autonym="Català" data-language-local-name="Katalanisch" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Konvergence_n%C3%A1hodn%C3%BDch_prom%C4%9Bnn%C3%BDch" title="Konvergence náhodných proměnných – Tschechisch" lang="cs" hreflang="cs" data-title="Konvergence náhodných proměnných" data-language-autonym="Čeština" data-language-local-name="Tschechisch" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%8D%CE%B3%CE%BA%CE%BB%CE%B9%CF%83%CE%B7_%CF%84%CF%85%CF%87%CE%B1%CE%AF%CF%89%CE%BD_%CE%BC%CE%B5%CF%84%CE%B1%CE%B2%CE%BB%CE%B7%CF%84%CF%8E%CE%BD" title="Σύγκλιση τυχαίων μεταβλητών – Griechisch" lang="el" hreflang="el" data-title="Σύγκλιση τυχαίων μεταβλητών" data-language-autonym="Ελληνικά" data-language-local-name="Griechisch" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Convergence_of_random_variables" title="Convergence of random variables – Englisch" lang="en" hreflang="en" data-title="Convergence of random variables" data-language-autonym="English" data-language-local-name="Englisch" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Convergencia_de_variables_aleatorias" title="Convergencia de variables aleatorias – Spanisch" lang="es" hreflang="es" data-title="Convergencia de variables aleatorias" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Konbergentzia_estokastiko" title="Konbergentzia estokastiko – Baskisch" lang="eu" hreflang="eu" data-title="Konbergentzia estokastiko" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%87%D9%85%DA%AF%D8%B1%D8%A7%DB%8C%DB%8C_%D9%85%D8%AA%D8%BA%DB%8C%D8%B1%D9%87%D8%A7%DB%8C_%D8%AA%D8%B5%D8%A7%D8%AF%D9%81%DB%8C" title="همگرایی متغیرهای تصادفی – Persisch" lang="fa" hreflang="fa" data-title="همگرایی متغیرهای تصادفی" data-language-autonym="فارسی" data-language-local-name="Persisch" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Convergence_de_variables_al%C3%A9atoires" title="Convergence de variables aléatoires – Französisch" lang="fr" hreflang="fr" data-title="Convergence de variables aléatoires" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%AA%D7%9B%D7%A0%D7%A1%D7%95%D7%AA_(%D7%94%D7%A1%D7%AA%D7%91%D7%A8%D7%95%D7%AA)" title="התכנסות (הסתברות) – Hebräisch" lang="he" hreflang="he" data-title="התכנסות (הסתברות)" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Convergenza_di_variabili_casuali" title="Convergenza di variabili casuali – Italienisch" lang="it" hreflang="it" data-title="Convergenza di variabili casuali" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%A2%BA%E7%8E%87%E5%A4%89%E6%95%B0%E3%81%AE%E5%8F%8E%E6%9D%9F" title="確率変数の収束 – Japanisch" lang="ja" hreflang="ja" data-title="確率変数の収束" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%99%95%EB%A5%A0_%EB%B3%80%EC%88%98%EC%9D%98_%EC%88%98%EB%A0%B4" title="확률 변수의 수렴 – Koreanisch" lang="ko" hreflang="ko" data-title="확률 변수의 수렴" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Convergentie_(kansrekening)" title="Convergentie (kansrekening) – Niederländisch" lang="nl" hreflang="nl" data-title="Convergentie (kansrekening)" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Zbie%C5%BCno%C5%9B%C4%87_wed%C5%82ug_rozk%C5%82adu" title="Zbieżność według rozkładu – Polnisch" lang="pl" hreflang="pl" data-title="Zbieżność według rozkładu" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Converg%C3%AAncia_de_vari%C3%A1veis_aleat%C3%B3rias" title="Convergência de variáveis aleatórias – Portugiesisch" lang="pt" hreflang="pt" data-title="Convergência de variáveis aleatórias" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D1%85%D0%BE%D0%B4%D0%B8%D0%BC%D0%BE%D1%81%D1%82%D1%8C_%D0%BF%D0%BE_%D1%80%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D1%8E" title="Сходимость по распределению – Russisch" lang="ru" hreflang="ru" data-title="Сходимость по распределению" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%97%D0%B1%D1%96%D0%B6%D0%BD%D1%96%D1%81%D1%82%D1%8C_%D0%B2%D0%B8%D0%BF%D0%B0%D0%B4%D0%BA%D0%BE%D0%B2%D0%B8%D1%85_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD" title="Збіжність випадкових величин – Ukrainisch" lang="uk" hreflang="uk" data-title="Збіжність випадкових величин" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%B1_h%E1%BB%99i_t%E1%BB%A5_c%E1%BB%A7a_c%C3%A1c_bi%E1%BA%BFn_ng%E1%BA%ABu_nhi%C3%AAn" title="Sự hội tụ của các biến ngẫu nhiên – Vietnamesisch" lang="vi" hreflang="vi" data-title="Sự hội tụ của các biến ngẫu nhiên" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a 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Konvergenz (Stochastik) – Wikipedia