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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,\dots are independent and identically-distributed random variables in \mathbb with common cumulative distribution function F(x). The ''empirical distribution function'' for X_1,\dots,X_n is defined by
:F_n(x)=\frac\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
:\|F_n - F\|_\infty = \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 (-\infty,x] by an arbitrary set ''C'' from a class of sets \mathcal to obtain an empirical measure indexed by sets C \in \mathcal.
:P_n(C)=\frac\sum_^n I_C(X_i), C\in\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\mapsto P_nf=\int_SfdP_n=\frac\sum_^n f(X_i), f\in\mathcal.
Then it becomes an important property of these classes that the strong law of large numbers holds uniformly on \mathcal or \mathcal.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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