## The Annals of Probability

### Uniform convergence of Vapnik–Chervonenkis classes under ergodic sampling

#### Abstract

We show that if $\mathcal{X}$ is a complete separable metric space and $\mathcal{C}$ is a countable family of Borel subsets of $\mathcal{X}$ with finite VC dimension, then, for every stationary ergodic process with values in $\mathcal{X}$, the relative frequencies of sets $C\in\mathcal{C}$ converge uniformly to their limiting probabilities. Beyond ergodicity, no assumptions are imposed on the sampling process, and no regularity conditions are imposed on the elements of $\mathcal{C}$. The result extends existing work of Vapnik and Chervonenkis, among others, who have studied uniform convergence for i.i.d. and strongly mixing processes. Our method of proof is new and direct: it does not rely on symmetrization techniques, probability inequalities or mixing conditions. The uniform convergence of relative frequencies for VC-major and VC-graph classes of functions under ergodic sampling is established as a corollary of the basic result for sets.

#### Article information

Source
Ann. Probab., Volume 38, Number 4 (2010), 1345-1367.

Dates
First available in Project Euclid: 8 July 2010

https://projecteuclid.org/euclid.aop/1278593952

Digital Object Identifier
doi:10.1214/09-AOP511

Mathematical Reviews number (MathSciNet)
MR2663629

Zentralblatt MATH identifier
1220.60019

#### Citation

Adams, Terrence M.; Nobel, Andrew B. Uniform convergence of Vapnik–Chervonenkis classes under ergodic sampling. Ann. Probab. 38 (2010), no. 4, 1345--1367. doi:10.1214/09-AOP511. https://projecteuclid.org/euclid.aop/1278593952

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