The Annals of Probability

The Glivenko-Cantelli Problem

Michel Talagrand

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Abstract

We give a new type of characterization of the Glivenko-Cantelli classes. In the case of a class $\mathscr{L}$ of sets, the characterization is closely related to the configuration that the sets of $\mathscr{L}$ can have. It allows one to decide simply whether a given class is a Glivenko-Cantelli class. The characterization is based on a new measure theoretic analysis of sets of measurable functions. This analysis also gives an approximation theorem for Glivenko-Cantelli classes, sharpenings of the Vapnik-Cervonenkis criteria and the value of the asymptotic discrepancy for classes that are not Glivenko-Cantelli. An application is given to the law of large numbers in a Banach space for functions that need not be random variables.

Article information

Source
Ann. Probab., Volume 15, Number 3 (1987), 837-870.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992069

Digital Object Identifier
doi:10.1214/aop/1176992069

Mathematical Reviews number (MathSciNet)
MR893902

Zentralblatt MATH identifier
0632.60024

JSTOR
links.jstor.org

Subjects
Primary: 60F15: Strong theorems
Secondary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence 28A51: Lifting theory [See also 46G15] 60F05: Central limit and other weak theorems

Keywords
Uniform law of large numbers empirical process empirical discrepancy Pettis norm

Citation

Talagrand, Michel. The Glivenko-Cantelli Problem. Ann. Probab. 15 (1987), no. 3, 837--870. doi:10.1214/aop/1176992069. https://projecteuclid.org/euclid.aop/1176992069


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