The central limit theorem for weighted empirical processes indexed by sets

Sufficient conditions are found for the weak convergence of a weighted empirical process {([nu]n(C)/q(P(C))) 1 [P(C) [succeeds, curly equals] [lambda]n]: C [set membership, variant] }, indexed by a class of sets and weighted by a function q of the size of each set. We find those functions q which allow weak convergence to a sample-continuous Gaussian process, and, given q, determine the fastest rate at which one may allow [lambda]n --> 0.