Glivenko-Cantelli class

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Consider a set S with a sigma algebra of Borel subsets A. For a class of subsets, {\mathcal C}=\{C:C\subset S\} and a probability measure P on (S,A) define a random variable (compare with empirical processes indexed by \mathcal{C} and KS statistics)

\|P_n-P\|_{\mathcal C}=\sup_{C\in {\mathcal C}} |P_n(C)-P(C)|

where P_n(C)=\frac 1n \sum_{i=1}^n I(X_i\in C) is empirical measure.

A class \mathcal C is called Glivenko-Cantelli class (or GC class) with respect to a probability measure P if any of the following equivalent statements is true

1. \|P_n-P\|_\mathcal{C}\to 0 almost surely as n\to\infty.
2. \|P_n-P\|_\mathcal{C}\to 0 in probability as n\to\infty.
3. \mathbb{E}\|P_n-P\|_\mathcal{C}\to 0, as n\to\infty (convergence in mean).

A class \mathcal C is called universal Glivenko-Cantelli class, if it is GC class with respect to any probability measure P on (S,A).

A class \mathcal C is called uniformly Glivenko-Cantelli class, if

\sup_{P\in \mathcal{P}(S)} \mathbb E \|P_n-P\|_\mathcal{C}\to 0 as n\to\infty.

where \mathcal{P}(S) is all probability measures on (S,A).

Example 1. Let S=\mathbb R and {\mathcal C}=\{(-\infty,t]:t\in {\mathbb R}\}. By the Donsker theorem, \|P_n-P\|_{\mathcal C} \sim n^{-1/2} for all P, that is \mathcal{C} is uniformly Glivenko-Cantelli class.

Example 2. Let P be a nonatomic probability measure on S and \mathcal{C} be a class of all finite subsets in S. Because A_n=\{X_1,\ldots,X_n\}\in \mathcal{C}, P(An) = 0, Pn(An) = 1, we have that \|P_n-P\|_{\mathcal C}=1 and so \mathcal{C} is not a GC class with respect to P.

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