Convergence of stochastic empirical measures

Abstract

Let Pn be a random probability measure on a metric space S. Let Pˆn be the empirical measure of kn iid random variables, each distributed according to Pn. Our main theorem asserts that if {Pn} converges in distribution, as random probability measures on S, then so does {Pˆn}. Applications of the result to the study of bootstrap and other stochastic procedures are given.