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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Proofs of convergence of random variables ： ウィキペディア英語版 Proofs of convergence of random variables This article is supplemental for “Convergence of random variables” and provides proofs for selected results. Several results will be established using the portmanteau lemma: A sequence converges in distribution to ''X'' if and only if any of the following conditions are met: E() → E() for all bounded, continuous functions ''f''; E() → E() for all bounded, Lipschitz functions ''f''; limsup ≤ Pr(''X'' ∈ ''C'') for all closed sets ''C''; == Convergence almost surely implies convergence in probability== : $X\_n\backslash \; \backslash xrightarrow\backslash \; X\; \backslash quad\backslash Rightarrow\backslash quad\; X\_n\backslash \; \backslash xrightarrow\backslash \; X$ Proof: If converges to ''X'' almost surely, it means that the set of points has measure zero; denote this set ''O''. Now fix ε > 0 and consider a sequence of sets : $A\_n\; =\; \backslash bigcup\_\; \backslash left\; \backslash$ This sequence of sets is decreasing: ''A''''n'' ⊇ ''A''''n''+1 ⊇ ..., and it decreases towards the set :$A\_\; =\; \backslash bigcap\_\; A\_n.$ For this decreasing sequence of events, their probabilities are also a decreasing sequence, and it decreases towards the Pr(''A''∞); we shall show now that this number is equal to zero. Now any point ω in the complement of ''O'' is such that lim ''Xn''(ω) = ''X''(ω), which implies that |''Xn''(ω) − ''X''(ω)| < ε for all ''n'' greater than a certain number ''N''. Therefore, for all ''n'' ≥ ''N'' the point ω will not belong to the set ''An'', and consequently it will not belong to ''A''∞. This means that ''A''∞ is disjoint with ''O'', or equivalently, ''A''∞ is a subset of ''O'' and therefore Pr(''A''∞) = 0. Finally, consider : $\backslash operatorname\backslash left(|X\_n-X|>\backslash varepsilon\backslash right)\; \backslash leq\; \backslash operatorname(A\_n)\; \backslash \; \backslash underset\; 0,$ which by definition means that ''Xn'' converges in probability to ''X''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Proofs of convergence of random variables」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.