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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 Kolmogorov's zero–one law ： ウィキペディア英語版 Kolmogorov's zero–one law In probability theory, Kolmogorov's zero–one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, called a ''tail event'', will either almost surely happen or almost surely not happen; that is, the probability of such an event occurring is zero or one. Tail events are defined in terms of infinite sequences of random variables. Suppose :$X\_1,X\_2,X\_3,\backslash dots\backslash ,$ is an infinite sequence of independent random variables (not necessarily identically distributed). Let $\backslash mathcal$ be the σ-algebra generated by the $X\_i$. Then, a tail event $F\; \backslash in\; \backslash mathcal$ is an event which is probabilistically independent of each finite subset of these random variables. (Note: $F$ belonging to $\backslash mathcal$ implies that membership in $F$ is uniquely determined by the values of the $X\_i$ but the latter condition is strictly weaker and does not suffice to prove the zero-one law.) For example, the event that the sequence converges, and the event that its sum converges are both tail events. In an infinite sequence of coin-tosses, a sequence of 100 consecutive heads occurring infinitely many times is a tail event. In many situations, it can be easy to apply Kolmogorov's zero–one law to show that some event has probability 0 or 1, but surprisingly hard to determine ''which'' of these two extreme values is the correct one. == Formulation == A more general statement of Kolmogorov's zero–one law holds for sequences of independent σ-algebras. Let (Ω,''F'',''P'') be a probability space and let ''F''''n'' be a sequence of mutually independent σ-algebras contained in ''F''. Let :$G\_n=\backslash sigma\backslash bigg(\backslash bigcup\_^\backslash infty\; F\_k\backslash bigg)$ be the smallest σ-algebra containing ''F''''n'', ''F''''n''+1, …. Then Kolmogorov's zero–one law asserts that for any event :$F\backslash in\; \backslash bigcap\_^\backslash infty\; G\_n$ one has either ''P''(''F'') = 0 or 1. The statement of the law in terms of random variables is obtained from the latter by taking each ''F''''n'' to be the σ-algebra generated by the random variable ''X''''n''. A tail event is then by definition an event which is measurable with respect to the σ-algebra generated by all ''X''''n'', but which is independent of any finite number of ''X''''n''. That is, a tail event is precisely an element of the intersection $\backslash textstyle$. 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「Kolmogorov's zero–one law」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.