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July 2010 Uniform convergence of Vapnik–Chervonenkis classes under ergodic sampling
Terrence M. Adams, Andrew B. Nobel
Ann. Probab. 38(4): 1345-1367 (July 2010). DOI: 10.1214/09-AOP511


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.


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Terrence M. Adams. Andrew B. Nobel. "Uniform convergence of Vapnik–Chervonenkis classes under ergodic sampling." Ann. Probab. 38 (4) 1345 - 1367, July 2010.


Published: July 2010
First available in Project Euclid: 8 July 2010

zbMATH: 1220.60019
MathSciNet: MR2663629
Digital Object Identifier: 10.1214/09-AOP511

Primary: 37A50 , 60C05 , 60F15
Secondary: 37A30 , 60G10

Keywords: ergodic process , Uniform convergence , uniform law of large numbers , VC class , VC dimension

Rights: Copyright © 2010 Institute of Mathematical Statistics

Vol.38 • No. 4 • July 2010
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