## Bernoulli

• Bernoulli
• Volume 18, Number 4 (2012), 1310-1319.

### Uniform approximation of Vapnik–Chervonenkis classes

#### 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.

#### Article information

Source
Bernoulli, Volume 18, Number 4 (2012), 1310-1319.

Dates
First available in Project Euclid: 12 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.bj/1352727812

Digital Object Identifier
doi:10.3150/11-BEJ379

Mathematical Reviews number (MathSciNet)
MR2995797

Zentralblatt MATH identifier
1268.60037

#### Citation

Adams, Terrence M.; Nobel, Andrew B. Uniform approximation of Vapnik–Chervonenkis classes. Bernoulli 18 (2012), no. 4, 1310--1319. doi:10.3150/11-BEJ379. https://projecteuclid.org/euclid.bj/1352727812

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