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November 2012 Uniform approximation of Vapnik–Chervonenkis classes
Terrence M. Adams, Andrew B. Nobel
Bernoulli 18(4): 1310-1319 (November 2012). DOI: 10.3150/11-BEJ379

Abstract

For any family of measurable sets in a probability space, we show that either (i) the family has infinite Vapnik–Chervonenkis (VC) dimension or (ii) for every $\varepsilon >0$ there is a finite partition $\pi$ such the essential $\pi$-boundary of each set has measure at most $\varepsilon $. Immediate corollaries include the fact that a separable family with finite VC dimension has finite bracketing numbers, and satisfies uniform laws of large numbers for every ergodic process. From these corollaries, we derive analogous results for VC major and VC graph families of functions.

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Terrence M. Adams. Andrew B. Nobel. "Uniform approximation of Vapnik–Chervonenkis classes." Bernoulli 18 (4) 1310 - 1319, November 2012. https://doi.org/10.3150/11-BEJ379

Information

Published: November 2012
First available in Project Euclid: 12 November 2012

zbMATH: 1268.60037
MathSciNet: MR2995797
Digital Object Identifier: 10.3150/11-BEJ379

Rights: Copyright © 2012 Bernoulli Society for Mathematical Statistics and Probability

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Vol.18 • No. 4 • November 2012
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