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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 Concentration of measure ： ウィキペディア英語版 Concentration of measure In mathematics, concentration of measure (about a median) is a principle that is applied in measure theory, probability and combinatorics, and has consequences for other fields such as Banach space theory. Informally, it states that "A random variable that depends in a Lipschitz way on many independent variables (but not too much on any of them) is essentially constant". 〔Michel Talagand, A New Look at Independence, The Annals of Probability, 1996, Vol. 24, No.1, 1-34〕 The c.o.m. phenomenon was put forth in the early 1970s by Vitali Milman in his works on the local theory of Banach spaces, extending an idea going back to the work of Paul Lévy.〔"''The concentration of $f\_\backslash ast(\backslash mu)$, ubiquitous in the probability theory and statistical mechanics, was brought to geometry (starting from Banach spaces) by Vitali Milman, following the earlier work by Paul Lévy''" - M. Gromov, Spaces and questions, GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. 2000, Special Volume, Part I, 118–161.〕〔"''The idea of concentration of measure (which was discovered by V.Milman) is arguably one of the great ideas of analysis in our times. While its impact on Probability is only a small part of the whole picture, this impact should not be ignored.''" - M. Talagrand, A new look at independence, Ann. Probab. 24 (1996), no. 1, 1–34.〕 It was further developed in the works of Milman and Gromov, Maurey, Pisier, Shechtman, Talagrand, Ledoux, and others. ==The general setting== Let $(X,\; d,\; \backslash mu)$ be a metric measure space, $\backslash mu(X)\; =\; 1$. Let :$\backslash alpha(\backslash epsilon)\; =\; \backslash sup\; \backslash left\backslash ,$ where :$A\_\backslash epsilon\; =\; \backslash left\backslash$ is the $\backslash epsilon$-''extension'' of a set $A$. The function $\backslash alpha(\backslash cdot)$ is called the ''concentration rate'' of the space $X$. The following equivalent definition has many applications: :$\backslash alpha(\backslash epsilon)\; =\; \backslash sup\; \backslash left\backslash )\; \backslash right\backslash \},$ where the supremum is over all 1-Lipschitz functions $F:\; X\; \backslash to\; \backslash mathbb$, and the median (or Levy mean) $M\; =\; \backslash mathop\; F$ is defined by the inequalities :$\backslash mu\; \backslash \; \backslash geq\; 1/2,\; \backslash ,\; \backslash mu\; \backslash \; \backslash geq\; 1/2.$ Informally, the space $X$ exhibits a concentration phenomenon if $\backslash alpha(\backslash epsilon)$ decays very fast as $\backslash epsilon$ grows. More formally, a family of metric measure spaces $(X\_n,\; d\_n,\; \backslash mu\_n)$ is called a ''Lévy family'' if the corresponding concentration rates $\backslash alpha\_n$ satisfy :$\backslash forall\; \backslash epsilon\; >\; 0\; \backslash ,\backslash ,\; \backslash alpha\_n(\backslash epsilon)\; \backslash to\; 0\; n\backslash to\; \backslash infty,$ and a ''normal Lévy family'' if :$\backslash forall\; \backslash epsilon\; >\; 0\; \backslash ,\backslash ,\; \backslash alpha\_n(\backslash epsilon)\; \backslash leq\; C\; \backslash exp(-c\; n\; \backslash epsilon^2)$ for some constants $c,C>0$. For examples see below. 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「Concentration of measure」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.