Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-15T10:18:53.941Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get accessRights & Permissions [Opens in a new window]

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

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 4 , December 1979, pp. 820 - 833
Copyright
Copyright © Applied Probability Trust 1979 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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.

References

Billingsley, P. and Topsøe, F. (1967) Uniformity in weak convergence. Z. Wahrscheinlichkeitsth. 7, 1–16.CrossRefGoogle Scholar
Bolthausen, E. (1978) Weak convergence of an empirical process indexed by the closed convex subsets of I 2 . Z. Wahrscheinlichkeitsth. 43, 173–181.Google Scholar
Dudley, R. M. (1978) Central limit theorems for empirical measures. Ann. Prob. 6, 899–929.Google Scholar
Eddy, W. and Hartigan, J. (1977) Uniform convergence of the empirical distribution function over convex sets. Ann. Statist. 5, 370–374.Google Scholar
Eggleston, H. G. (1969) Convexity. Cambridge University Press.Google Scholar
Elker, J. (1975) Über ein gleichmäßiges Gesetz der großen Zahlen. Unpublished Diplomarbeit, Institut für Mathematik, Ruhr-Universität Bochum.Google Scholar
Gänßler, P. and Stute, W. (1979) Empirical processes: a survey on some results in the i.i.d. case. Ann. Prob. 7, 193–243.Google Scholar
Ranga Rao, R. (1962) Relations between weak and uniform convergence of measures with applications. Ann. Math. Statist. 33, 659–680.Google Scholar
Sager, T. W. (1978) Estimation of a multivariate mode. Ann. Statist. 6, 802–812.CrossRefGoogle Scholar
Steele, J. M. (1978) Empirical discrepancies and subadditive processes. Ann. Prob. 6, 118–127.CrossRefGoogle Scholar
Stute, W. (1976) On a generalization of the Glivenko–Cantelli theorem. Z. Wahrscheinlichkeitsth. 35, 167–175.Google Scholar
Stute, W. (1977) Convergence rates for the isotrope discrepancy. Ann. Prob. 5, 707–723.CrossRefGoogle Scholar
Topsøe, F. (1970) On the Glivenko–Cantelli theorem. Z. Wahrscheinlichkeitsth. 14, 239–250.CrossRefGoogle Scholar
Topsøe, F. (1977) Uniformity in convergence of measures. Z. Wahrscheinlichkeitsth. 39, 1–30.Google Scholar
Vapnik, V. N. and Červonenkis, A. Ya. (1971) On the uniform convergence of relative frequencies of events to their probabilities. Theory Prob. Appl. 16, 264–280.Google Scholar
Wolfowitz, J. (1954) Generalisations of the theorem of Glivenko–Cantelli. Ann. Math. Statist. 25, 131–138.CrossRefGoogle Scholar