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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 Prokhorov's theorem ： ウィキペディア英語版 Prokhorov's theorem In measure theory Prokhorov’s theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures. It is credited to the Soviet mathematician Yuri Vasilyevich Prokhorov, who considered probability measures on complete separable metric spaces. The term "Prokhorov’s theorem" is also applied to later generalizations to either the direct or the inverse statements. ==Statement of the theorem== Let $(S,\; \backslash rho)$ be a separable metric space. Let $\backslash mathcal(S)$ denote the collection of all probability measures defined on $S$ (with its Borel σ-algebra). Theorem. # A collection $K\backslash subset\; \backslash mathcal(S)$ of probability measures is tight if and only if the closure of $K$ is sequentially compact in the space $\backslash mathcal(S)$ equipped with the topology of weak convergence. # The space $\backslash mathcal(S)$ with the topology of weak convergence is metrizable. # Suppose that in addition, $(S,\backslash rho)$ is a complete metric space (so that $(S,\backslash rho)$ is a Polish space). There is a complete metric $d\_0$ on $\backslash mathcal(S)$ equivalent to the topology of weak convergence; moreover, $K\backslash subset\; \backslash mathcal(S)$ is tight if and only if the closure of $K$ in $(\backslash mathcal(S),d\_0)$ is compact. 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「Prokhorov's 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.