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

TL;DR

This paper investigates the asymptotic behavior of weighted bootstrap methods for u- and v-statistics, establishing their consistency and deriving a conditional CLT under broad conditions.

Contribution

It extends existing results on the consistency of u- and v-statistics to weighted bootstrap settings for both i.i.d. and non-i.i.d. data, introducing a new conditioning approach.

Findings

Proves the consistency of weighted bootstrap u- and v-statistics.

Derives a conditional CLT for these bootstrap statistics.

Extends classical results to more general weighted and dependent data scenarios.

Abstract

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.