The Annals of Probability

Central Limit Theorems for Empirical Measures

R. M. Dudley

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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 $r, \sigma_m(P(C_m)(1 - P(C_m)))^r < \infty$.

Article information

Ann. Probab., Volume 6, Number 6 (1978), 899-929.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 60F05: Central limit and other weak theorems
Secondary: 60B10: Convergence of probability measures 60G17: Sample path properties 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05] 28A40

Central limit theorems empirical measures Donsker classes Effros Borel structure metric entropy with inclusion two-sample case Vapnik-Cervonenkis classes


Dudley, R. M. Central Limit Theorems for Empirical Measures. Ann. Probab. 6 (1978), no. 6, 899--929. doi:10.1214/aop/1176995384.

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  • See Correction: R. M. Dudley. Corrections to "Central Limit Theorems for Empirical Measures". Ann. Probab., Vol. 7, Iss. 5 (1979), 909--911.