Large Deviations Bounds for Empirical Processes
Vapnik-Chervonenkis bounds on speeds of convergence of empirical means to their expectations have been continuously improved over the years. The result obtained by M. Talagrand in 1994 11] seems to provide the nal word as far as universal bounds are concerned. However, for xed families of underlying probability distributions, the exponential rate in the deviation term can be fairly improved by the more adequate Cram er transform, as shown by large deviations theorems. Dans ce qui suit, on note ? une famille d'ensembles mesurables C et V la VC dimension de ?. On rappelle que V est un indice de complexit e combinatoire qui caract erise la capacit e de la famille d'ensembles a pulv eriser un ensemble de points. n variables al eatoires sur (R d ; B), ind ependantes et identiquement distribu ees, de m^ eme loi .