Uniform and universal Glivenko-Cantelli classes
type: Item
instance of: scholarly article
title: Uniform and universal Glivenko-Cantelli classes
zbMATH Open document ID: 0732.60035
DOI: 10.1007/BF01210321
author: Joel Zinn
author: Richard M. Dudley
author: Evarist Giné
published in: Journal of Theoretical Probability
publication date: 1991-01-01T00:00:00Z
review text: A class \(\tilde F\) of measurable functions on a probability space is called a Glivenko-Cantelli class if the empirical measures \(P_ n\) converge to the true P uniformly over \(\tilde F\) almost surely. \(\tilde F\) is a universal Glivenko-Cantelli class if it is a Glivenko-Cantelli class for all laws P on a measurable space, and a uniform Glivenko- Cantelli class if the convergence is also uniform in P. The authors give general sufficient conditions for the Glivenko-Cantelli and universal Glivenko-Cantelli properties and examples to show that some stronger conditions are not necessary. The uniform Glivenko-Cantelli property is characterized, under measurability assumptions, by an entropy condition.
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 4211167
zbMATH Keywords: law of large numbers
zbMATH Keywords: Vapnik-Červonenkis classes
zbMATH Keywords: empirical measures
zbMATH Keywords: Glivenko-Cantelli class
zbMATH Keywords: entropy condition
reviewed by: Zdzisław Rychlik
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4211167