Open Access
July, 1987 The Glivenko-Cantelli Problem
Michel Talagrand
Ann. Probab. 15(3): 837-870 (July, 1987). DOI: 10.1214/aop/1176992069


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.


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Michel Talagrand. "The Glivenko-Cantelli Problem." Ann. Probab. 15 (3) 837 - 870, July, 1987.


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

zbMATH: 0632.60024
MathSciNet: MR893902
Digital Object Identifier: 10.1214/aop/1176992069

Primary: 60F15
Secondary: 28A20 , 28A51 , 60B12 , 60F05

Keywords: empirical discrepancy , empirical process , Pettis norm , uniform law of large numbers

Rights: Copyright © 1987 Institute of Mathematical Statistics

Vol.15 • No. 3 • July, 1987
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