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October, 1987 Universal Donsker Classes and Metric Entropy
R. M. Dudley
Ann. Probab. 15(4): 1306-1326 (October, 1987). DOI: 10.1214/aop/1176991978


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.


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R. M. Dudley. "Universal Donsker Classes and Metric Entropy." Ann. Probab. 15 (4) 1306 - 1326, October, 1987.


Published: October, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0631.60004
MathSciNet: MR905333
Digital Object Identifier: 10.1214/aop/1176991978

Primary: 60F17
Secondary: 60F05 , 60G17 , 60G20

Keywords: central limit theorems , Vapnik-Cervonenkis classes

Rights: Copyright © 1987 Institute of Mathematical Statistics

Vol.15 • No. 4 • October, 1987
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