Rates of growth and sample moduli for weighted empirical processes indexed by sets

Abstract

Probability inequalities are obtained for the supremum of a weighted empirical process indexed by a Vapnik-Červonenkis class C of sets. These inequalities are particularly useful under the assumption P(∪{C∈C:P(C)<t})»0 as t»0. They are used to obtain almost sure bounds on the rate of growth of the process as the sample size approaches infinity, to find an asymptotic sample modulus for the unweighted empirical process, and to study the ratio P _n/P of the empirical measure to the actual measure.