Asymptotics of randomly weighted uu- and vv-statistics: Application to bootstrap

This paper is mainly concerned with asymptotic studies of weighted bootstrap for uu- and vv-statistics. We derive the consistency of the weighted bootstrap uu- and vv-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 uu-statistics and also consistency of sums of arrays of random variables. We also employ a new approach to conditioning to derive a conditional central limit theorem (CLT) for weighted bootstrap uu- and vv-statistics, assuming the same conditions as the classical CLT for regular uu- and vv-statistics.