Asymptotic Distributions for Weighted $U$-Statistics

Abstract

We prove limit theorems for weighted $U$-statistics and express the limit by means of multiple stochastic integrals. This is a generalization of the paper by O'Neil and Redner. In that paper the method of moments was applied which does not work in the general case. Hence we had to work out a different method. In particular, in Theorem 4 we describe the limit of a model proposed by O'Neil and Redner. In this model the weight functions cause an intricate cancellation, and the limit can be presented as a sum of multiple stochastic integrals with different multiplicities.