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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 Glivenko–Cantelli theorem ： ウィキペディア英語版 Glivenko–Cantelli theorem In the theory of probability, the Glivenko–Cantelli theorem, named after Valery Ivanovich Glivenko and Francesco Paolo Cantelli, determines the asymptotic behaviour of the empirical distribution function as the number of independent and identically distributed observations grows. The uniform convergence of more general empirical measures becomes an important property of the Glivenko–Cantelli classes of functions or sets. The Glivenko–Cantelli classes arise in Vapnik–Chervonenkis theory, with applications to machine learning. Applications can be found in econometrics making use of M-estimators. Assume that $X\_1,X\_2,\backslash dots$ are independent and identically-distributed random variables in $\backslash mathbb$ with common cumulative distribution function $F(x)$. The ''empirical distribution function'' for $X\_1,\backslash dots,X\_n$ is defined by :$F\_n(x)=\backslash frac\backslash sum\_^n\; I\_(X\_i),$ where $I\_C$ is the indicator function of the set $C$. For every (fixed) $x$, $F\_n(x)$ is a sequence of random variables which converge to $F(x)$ almost surely by the strong law of large numbers, that is, $F\_n$ converges to $F$ pointwise. Glivenko and Cantelli strengthened this result by proving uniform convergence of $F\_n$ to $F$. Theorem :$\backslash |F\_n\; -\; F\backslash |\_\backslash infty\; =\; \backslash sup\_\; 0$ almost surely. This theorem originates with Valery Glivenko,〔Glivenko, V. (1933). Sulla determinazione empirica della legge di probabilita. Giorn. Ist. Ital. Attuari 4, 92-99.〕 and Francesco Cantelli,〔Cantelli, F. P. (1933). Sulla determinazione empirica delle leggi di probabilita. Giorn. Ist. Ital. Attuari 4, 221-424.〕 in 1933. Remarks *If $X\_n$ is a stationary ergodic process, then $F\_n(x)$ converges almost surely to $F(x)=E(1\_)$. The Glivenko–Cantelli theorem gives a stronger mode of convergence than this in the iid case. *An even stronger uniform convergence result for the empirical distribution function is available in the form of an extended type of law of the iterated logarithm. See asymptotic properties of the Empirical distribution function for this and related results. ==Empirical measures== One can generalize the ''empirical distribution function'' by replacing the set $(-\backslash infty,x]$ by an arbitrary set ''C'' from a class of sets $\backslash mathcal$ to obtain an empirical measure indexed by sets $C\; \backslash in\; \backslash mathcal.$ :$P\_n(C)=\backslash frac\backslash sum\_^n\; I\_C(X\_i),\; C\backslash in\backslash mathcal$ Where $I\_C(x)$ is the indicator function of each set $C$. Further generalization is the map induced by $P\_n$ on measurable real-valued functions ''f'', which is given by :$f\backslash mapsto\; P\_nf=\backslash int\_SfdP\_n=\backslash frac\backslash sum\_^n\; f(X\_i),\; f\backslash in\backslash mathcal.$ Then it becomes an important property of these classes that the strong law of large numbers holds uniformly on $\backslash mathcal$ or $\backslash mathcal$. 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「Glivenko–Cantelli 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.