The universal Glivenko-Cantelli property

Ramon van Handel

doi:10.1007/s00440-012-0416-5

The universal Glivenko-Cantelli property

Ramon van Handel

Research output: Contribution to journal › Article › peer-review

21 Scopus citations

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 language	English (US)
Pages (from-to)	911-934
Number of pages	24
Journal	Probability Theory and Related Fields
Volume	155
Issue number	3-4
DOIs	https://doi.org/10.1007/s00440-012-0416-5
State	Published - Apr 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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10.1007/s00440-012-0416-5

Cite this

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title = "The universal Glivenko-Cantelli property",

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.",

keywords = "Boolean independence, Entropy with bracketing, Uniform convergence of random measures, Uniformity classes, Universal Glivenko-Cantelli classes",

author = "{van Handel}, Ramon",

note = "Funding Information: This work was partially supported by NSF grant DMS-1005575.",

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doi = "10.1007/s00440-012-0416-5",

language = "English (US)",

volume = "155",

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T1 - The universal Glivenko-Cantelli property

AU - van Handel, Ramon

N1 - Funding Information: This work was partially supported by NSF grant DMS-1005575.

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N2 - 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.

AB - 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.

KW - Boolean independence

KW - Entropy with bracketing

KW - Uniform convergence of random measures

KW - Uniformity classes

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The universal Glivenko-Cantelli property

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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