The Annals of Probability

Universal Donsker Classes and Metric Entropy

R. M. Dudley

Full-text: Open access


Let $(X, \mathscr{A})$ be a measurable space and $\mathscr{F}$ a class of measurable functions on $X. \mathscr{F}$ is called a universal Donsker class if for every probability measure $P$ on $\mathscr{A}$, the centered and normalized empirical measures $n^{1/2}(P_n - P)$ converge in law, with respect to uniform convergence over $\mathscr{F}$, to the limiting "Brownian bridge" process $G_P$. Then up to additive constants, $\mathscr{F}$ must be uniformly bounded. Several nonequivalent conditions are shown to imply the universal Donsker property. Some are connected with the Vapnik-Cervonenkis combinatorial condition on classes of sets, others with metric entropy. The implications between the various conditions are considered. Bounds are given for the metric entropy of convex hulls in Hilbert space.

Article information

Ann. Probab., Volume 15, Number 4 (1987), 1306-1326.

First available in Project Euclid: 19 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60F05: Central limit and other weak theorems 60G17: Sample path properties 60G20: Generalized stochastic processes

Central limit theorems Vapnik-Cervonenkis classes


Dudley, R. M. Universal Donsker Classes and Metric Entropy. Ann. Probab. 15 (1987), no. 4, 1306--1326. doi:10.1214/aop/1176991978.

Export citation