Uniform convergence of Vapnik–Chervonenkis classes under ergodic sampling

@article{Adams2010UniformCO,
title={Uniform convergence of Vapnik–Chervonenkis classes under ergodic sampling},
author={Terrence M. Adams and Andrew B. Nobel},
journal={Annals of Probability},
year={2010},
volume={38},
pages={1345-1367}
}
• Published 1 July 2010
• Mathematics
• Annals of Probability
We show that if X is a complete separable metric space and C is a countable family of Borel subsets of X with finite VC dimension, then, for every stationary ergodic process with values in X, the relative frequencies of sets C ∈ 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 C. The result extends existing work of Vapnik and Chervonenkis, among others, who…
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