Summary
Letx 1,...,x n be independent random variables with uniform distribution over [0, 1]d, andX(n) be the centered and normalized empirical process associated tox 1,...,x n . Given a Vapnik-Chervonenkis classL of bounded functions from [0, 1]d intoR of bounded variation, we apply the one-dimensional dyadic scheme of Komlós, Major and Tusnády to get the best possible rate in Dudley's uniform central limit theorem for the empirical process {E (n)(h):h∈L}. WhenL fulfills some extra condition, we prove there exists some sequenceB n of Brownian bridges indexed byL such that
whereK (L) denotes the maximal variation of the elements ofL. This result is then applied to maximal deviations distributions for kernel density estimators under minimal assumptions on the sequence of bandwith parameters. We also derive some results concerning strong approximations for empirical processes indexed by classes of sets with uniformly small perimeter. For example, it follows from Beck's paper that the above result is optimal, up to a possible factor\(\sqrt {\log n}\), whenL is the class of Euclidean balls with radius less thanr.
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