The universal Glivenko-Cantelli property

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Abstract

Let F be a separable uniformly bounded family of measurable functions on a standard measurable space (X, X), and let N[] (F, ε, μ) be the smallest number of ε-brackets in L1(μ) needed to cover F. The following are equivalent: 1. F is a universal Glivenko-Cantelli class.2. N[](F, ε, μ) < ∞ for every ε < 0 and every probability measure μ 3. F is totally bounded in L1(μ) for every probability measure μ.4. F does not contain a Boolean σ-independent sequence. It follows that universal Glivenko-Cantelli classes are uniformity classes for general sequences of almost surely convergent random measures.

Original languageEnglish (US)
Pages (from-to)911-934
Number of pages24
JournalProbability Theory and Related Fields
Volume155
Issue number3-4
DOIs
StatePublished - Jan 1 2013

All Science Journal Classification (ASJC) codes

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Boolean independence
  • Entropy with bracketing
  • Uniform convergence of random measures
  • Uniformity classes
  • Universal Glivenko-Cantelli classes

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