Open Access
December, 1978 Central Limit Theorems for Empirical Measures
R. M. Dudley
Ann. Probab. 6(6): 899-929 (December, 1978). DOI: 10.1214/aop/1176995384


Let $(X, \mathscr{A}, P)$ be a probability space. Let $X_1, X_2,\cdots,$ be independent $X$-valued random variables with distribution $P$. Let $P_n := n^{-1}(\delta_{X_1} + \cdots + \delta_{X_n})$ be the empirical measure and let $\nu_n := n^\frac{1}{2}(P_n - P)$. Given a class $\mathscr{C} \subset \mathscr{a}$, we study the convergence in law of $\nu_n$, as a stochastic process indexed by $\mathscr{C}$, to a certain Gaussian process indexed by $\mathscr{C}$. If convergence holds with respect to the supremum norm $\sup_{C \in \mathscr{C}}|f(C)|$, in a suitable (usually nonseparable) function space, we call $\mathscr{C}$ a Donsker class. For measurability, $X$ may be a complete separable metric space, $\mathscr{a} =$ Borel sets, and $\mathscr{C}$ a suitable collection of closed sets or open sets. Then for the Donsker property it suffices that for some $m$, and every set $F \subset X$ with $m$ elements, $\mathscr{C}$ does not cut all subsets of $F$ (Vapnik-Cervonenkis classes). Another sufficient condition is based on metric entropy with inclusion. If $\mathscr{C}$ is a sequence $\{C_m\}$ independent for $P$, then $\mathscr{C}$ is a Donsker class if and only if for some $r, \sigma_m(P(C_m)(1 - P(C_m)))^r < \infty$.


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R. M. Dudley. "Central Limit Theorems for Empirical Measures." Ann. Probab. 6 (6) 899 - 929, December, 1978.


Published: December, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0404.60016
MathSciNet: MR512411
Digital Object Identifier: 10.1214/aop/1176995384

Primary: 60F05
Secondary: 28A05 , 28A40 , 60B10 , 60G17

Keywords: central limit theorems , Donsker classes , Effros Borel structure , empirical measures , metric entropy with inclusion , two-sample case , Vapnik-Cervonenkis classes

Rights: Copyright © 1978 Institute of Mathematical Statistics

Vol.6 • No. 6 • December, 1978
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