Read e-book online Wahrscheinlichkeitstheorie PDF

By Achim Klenke

ISBN-10: 3540255451

ISBN-13: 9783540255451

Dieses Lehrbuch bietet eine umfassende moderne Einführung in die wichtigsten Gebiete der Wahrscheinlichkeitstheorie und ihre maßtheoretischen Grundlagen. Themenschwerpunkte sind: Maß- und Integrationstheorie, Grenzwertsätze für Summen von Zufallsvariablen (Gesetze der Großen Zahl, Zentraler Grenzwertsatz, Ergodensätze, Gesetz vom iterierten Logarithmus, Invarianzprinzipien, unbegrenzt teilbare Verteilungen), Martingale, Perkolation, Markovketten und elektrische Netzwerke, Konstruktion stochastischer Prozesse, Poisson'scher Punktprozess, Brown'sche Bewegung, stochastisches vital und stochastische Differentialgleichungen. Mehr als 250 Übungsaufgaben und zahlreiche Abbildungen runden die Darstellung ab. Breite und Auswahl der Themen sind einmalig in der deutschsprachigen Literatur.

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Sei E eine endliche, nichtleere Menge und Ω = E N sowie (pe )e∈E ein Wahrscheinlichkeitsvektor. Dann gibt es ein eindeutig bestimmtes W-Maß μ auf σ(A) = B(Ω) mit n p ωi μ([ω1 , . . , ωn ]) = i=1 f¨ur alle ω1 , . . , ωn ∈ E und n ∈ N. Wir nennen μ das Produktmaß oder Bernoulli-Maß auf Ω mit Gewichten (pe )e∈E . ⊗N := μ. Wir schreiben auch e∈E pe δe E ⊗N Ferner nennen wir (2 ) := σ(A) die Produkt-σ-Algebra auf Ω. Auf Produktmaße gehen wir systematisch noch einmal in Kapitel 14 ein. Der Fortsetzungssatz liefert uns einen abstrakten Existenz- und Eindeutigkeitssatz f¨ur Maße, die wir zuvor nur auf einem Semiring A definiert hatten.

3 Fortsetzung von Maßen 25 n i=1 (bi − ai ) zu einem Maß auf der Borel’schen σ-Algebra B(Rn ) fortsetzen. 53 zu pr¨ufen, m¨ussen wir nur noch zeigen, dass μ σ-subadditiv ist. Seien also (a, b], (a(1), b(1)], (a(2), b(2)], . . ∈ A mit (a, b] ⊂ ∞ (a(k), b(k)]. 12) μ((a(k), b(k)]). 12) auf die endliche Additivit¨at zur¨uck zu f¨uhren. Sei also ε > 0, und sei f¨ur jedes k ∈ N ein bε (k) > b(k) so gew¨ahlt, dass μ((a(k), bε (k)]) ≤ μ((a(k), b(k)]) + ε 2−k−1 . Ferner sei aε ∈ (a, b) so gew¨ahlt, dass μ((aε , b]) ≥ μ((a, b]) − 2ε .

Ist A ∈ B(Rn ) mit n-dimensionalem LebesgueMaß λn (A) ∈ (0, ∞), so wird durch μ(B) := λn (B) λn (A) f¨ur B ∈ B(Rn ), B ⊂ A, ein W-Maß auf B(Rn ) definiert. Wir nennen μ die uniforme Verteilung oder A Gleichverteilung auf A und schreiben UA := μ. 1. 58(iv): Ein ∞ Maß n=1 αn δxn ist genau dann ein Lebesgue-Stieltjes Maß zu einer geeigneten ♣ Funktion F , wenn n: |xn |≤K αn < ∞ f¨ur jedes K > 0 gilt. 2. Sei Ω eine u¨ berabz¨ahlbare Menge und ω0 ∈ Ω ein beliebiges Element. Sei A = σ({ω} : ω ∈ {ω0 }).

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