Asymptotics of Randomly Weighted u- and v-statistics: Application to Bootstrap

This paper is mainly concerned with asymptotic studies of weighted bootstrap for u- and v-statistics. We derive the consistency of the weighted bootstrap u- and v-statistics, based on i.i.d. and non i.i.d. observations, from some more general results which we first establish for sums of randomly weighted arrays of random variables. Some of the results in this paper significantly extend some well-known results on consistency of u-statistics and also consistency of sums of arrays of random variables. We also employ a new approach to conditioning to derive a conditional CLT for weighted bootstrap u- and v-statistics, assuming the same conditions as the classical central limit theorems for regular u- and v-statistics.