Abstract
Let \({\mathcal{F}}\) be a separable uniformly bounded family of measurable functions on a standard measurable space \({(X, \mathcal{X})}\), and let \({N_{[]}(\mathcal{F}, \varepsilon, \mu)}\) be the smallest number of \({\varepsilon}\) -brackets in L 1(μ) needed to cover \({\mathcal{F}}\). The following are equivalent:
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1.
\({\mathcal{F}}\) is a universal Glivenko–Cantelli class.
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2.
\({N_{[]}(\mathcal{F},\varepsilon,\mu) < \infty}\) for every \({\varepsilon > 0}\) and every probability measure μ.
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3.
\({\mathcal{F}}\) is totally bounded in L 1(μ) for every probability measure μ.
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4.
\({\mathcal{F}}\) 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.
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This work was partially supported by NSF grant DMS-1005575.
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van Handel, R. The universal Glivenko–Cantelli property. Probab. Theory Relat. Fields 155, 911–934 (2013). https://doi.org/10.1007/s00440-012-0416-5
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DOI: https://doi.org/10.1007/s00440-012-0416-5
Keywords
- Universal Glivenko–Cantelli classes
- Uniformity classes
- Uniform convergence of random measures
- Entropy with bracketing
- Boolean independence