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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 Hsu–Robbins–Erdős theorem ： ウィキペディア英語版 Hsu–Robbins–Erdős theorem In the mathematical theory of probability, the Hsu–Robbins–Erdős theorem states that if $X\_1,\; \backslash ldots\; ,X\_n$ is a sequence of i.i.d. random variables with zero mean and finite variance and : $S\_n\; =\; X\_1\; +\; \backslash cdots\; +\; X\_n,\; \backslash ,$ then : for every $\backslash varepsilon\; >\; 0$. The result was proved by Pao-Lu Hsu and Herbert Robbins in 1947. This is an interesting strengthening of the classical strong law of large numbers in the direction of the Borel–Cantelli lemma. The idea of such a result is probably due to Robbins, but the method of proof is vintage Hsu.〔Chung, K. L. (1979). Hsu's work in probability. The Annals of Statistics, 479–483.〕 Hsu and Robbins further conjectured in 〔Hsu, P. L., & Robbins, H. (1947). Complete convergence and the law of large numbers. Proceedings of the National Academy of Sciences of the United States of America, 33(2), 25.〕 that the condition of finiteness of the variance of $X$ is also a necessary condition for to hold. Two years later, the famed mathematician Paul Erdős proved the conjecture.〔Erdos, P. (1949). On a theorem of Hsu and Robbins. The Annals of Mathematical Statistics, 286–291.〕 Since then, many authors extended this result in several directions.〔(Hsu-Robbins theorem for the correlated sequences )〕 ==References== 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「Hsu–Robbins–Erdős theorem」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.