  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.
