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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 Borel–Cantelli lemma ： ウィキペディア英語版 Borel–Cantelli lemma In probability theory, the Borel–Cantelli lemma is a theorem about sequences of events. In general, it is a result in measure theory. It is named after Émile Borel and Francesco Paolo Cantelli, who gave statement to the lemma in the first decades of the 20th century.〔E. Borel, "Les probabilités dénombrables et leurs applications arithmetiques" ''Rend. Circ. Mat. Palermo'' (2) 27 (1909) pp. 247–271.〕〔F.P. Cantelli, "Sulla probabilità come limite della frequenza", ''Atti Accad. Naz. Lincei'' 26:1 (1917) pp.39–45.〕 A related result, sometimes called the second Borel–Cantelli lemma, is a partial converse of the first Borel–Cantelli lemma. The lemma states that, under certain conditions, an event will either occur with probability zero or with probability one. As such, it is the best-known of a class of similar theorems, known as zero-one laws. Other examples include the Kolmogorov 0-1 law and the Hewitt–Savage zero-one law. ==Statement of lemma for probability spaces== Let ''E''''1'',''E''''2''...''E''''n'' be a sequence of events in some probability space. The Borel–Cantelli lemma states: :If the sum of the probabilities of the ''E''''n'' is finite :: :then the probability that infinitely many of them occur is 0, that is, ::$\backslash Pr\backslash left(\backslash limsup\_\; E\_n\backslash right)\; =\; 0.\backslash ,$ Here, "lim sup" denotes limit supremum of the sequence of events, and each event is a set of outcomes. That is, lim sup ''E''''n'' is the set of outcomes that occur infinitely many times within the infinite sequence of events (''E''''n''). Explicitly, :$\backslash limsup\_\; E\_n\; =\; \backslash bigcap\_^\; \backslash bigcup\_^\; E\_k.$ The theorem therefore asserts that if the sum of the probabilities of the events ''E''''n'' is finite, then the set of all outcomes that are "repeated" infinitely many times must occur with probability zero. Note that no assumption of independence is required. 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「Borel–Cantelli lemma」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.