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Glivenko–Cantelli theorems for classes of convex sets

Published online by Cambridge University Press:  01 July 2016

J. Elker ,
D. Pollard  and
W. Stute
J. Elker*
Affiliation:
Ruhr-Universität Bochum
D. Pollard
Affiliation:
Yale University
W. Stute
Affiliation:
Ruhr-Universität Bochum
*
∗Postal address: Institut für Mathematik, Ruhr-Universität Bochum, Universitätsstr. 150, GEB NA, Postfach 2148, 463 Bochum, West Germany.
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Abstract

Let denote the empirical distribution obtained from a sequence of i.i.d. -valued random vectors with common distribution P. If is a class of Borel subsets of then we say that it forms a Glivenko–Cantelli class for P if In this paper we describe a simple technique for identifying such classes, based on the idea of uniformity classes for setwise convergence. Classes for which the method proves successful include the closed half-spaces, closed balls, and the class of all convex subsets of .

Keywords

EMPIRICAL DISTRIBUTIONGLIVENKO-CANTELLI THEOREMUNIFORMITY CLASSES OF CONVEX SETS
Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 4 , December 1979 , pp. 820 - 833
Copyright
Copyright © Applied Probability Trust 1979 

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Footnotes

∗∗

Present address: Department of Statistics, Yale University, Box 2179, Yale Station, New Haven, CT 06520, U.S.A. Supported by a fellowship of the Alexander von Humboldt Foundation while visiting the Ruhr-Universität Bochum.

∗∗∗

Present address: Mathematisches Institut der Universität, Theresienstr. 39. D-8000 München 2, West Germany.

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