Mathematics – Probability
Scientific paper
2010-10-15
Annals of Probability 2010, Vol. 38, No. 4, 1345-1367
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/09-AOP511 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Scientific paper
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.
Adams Terrence M.
Nobel Andrew B.
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