The Glivenko-Cantelli problem, ten years later
type: Item
instance of: scholarly article
title: The Glivenko-Cantelli problem, ten years later
zbMATH Open document ID: 0880.60023
DOI: 10.1007/BF02214655
author: Michel Talagrand
published in: Journal of Theoretical Probability
publication date: 1997-11-06T00:00:00Z
review text: Loosely speaking, a Glivenko-Cantelli (GC) class \(Z\) consists of measurable uniformly bounded functions on a complete probability space \((\Omega, {\mathcal F}, {\mathbf P})\) s.t., given a sequence of \(\Omega\)-valued independent r.v.'s \(X_k\) of law \({\mathbf P}\), one has \[ \lim_{n\to\infty}{\mathbf E}\sup \Biggl\{\frac{1}{n}\Bigl|\sum_{k\leq n}(f(X_k)-{\mathbf E}f)\Bigr|,\;f\in Z\Biggr\}=0. \] Various results related to characterisation of GC classes are discussed. The author has simplified his previous difficult proofs obtained ten years ago [Ann. Probab. 15, 837-870 (1987; Zbl 0632.60024)]. The accent is made on the cases where involved analysis of measurability problems can be avoided.
reviewed by: Alexander V. Bulinski
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 1000174
zbMATH Keywords: Glivenko-Cantelli theorem
zbMATH Keywords: strong law of large numbers
zbMATH Keywords: Vapnik-Chervonenkis condition
zbMATH Keywords: Glivenko-Cantelli class
zbMATH Keywords: measurability problems
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1000174