In this paper we give optimal constants in Talagrand's concentration inequalities for maxima of empirical processes associated to independent and eventually nonidentically distributed random variables. Our approach is based on the entropy method introduced by Ledoux.Comment: Published at http://dx.doi.org/10.1214/009117905000000044 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Let Xl, 9 9 xn be independent random variables with uniform distribution over [0, 1] d, and X (") be the centered and normalized empirical process associated to xl, ..., xn. Given a Vapnik-Chervonenkis class 5 p of bounded functions from [0, 1] d into IR of bounded variation, we apply the one-dimensional dyadic scheme of Komlds, Major and Tusnfidy to get the best possible rate in Dudley's uniform central limit theorem for the empirical process {X(n~(h):h E 5 ~ }. When 5 p fulfills some extra condition, we prove there exists some sequence B,, of Brownian bridges indexed by ~ such that sup iX(")(h) -B.(h)[ = O(n-1/Zlogn v n 1/(2d~x/K(SP)logn)a.s.h~5 owhere K (5 P) denotes the maximal variation of the elements of 5 P. 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 lx/~n, when 5 P is the class of Euclidean balls with radius less than r.