Degenerate U- and V-statistics under weak dependence: Asymptotic theory and bootstrap consistency

TL;DR

This paper establishes the asymptotic behavior and bootstrap consistency of degenerate U- and V-statistics under weak dependence, providing theoretical foundations for their use in hypothesis testing.

Contribution

It derives the limit distributions of degenerate U- and V-statistics under weak dependence and proves bootstrap methods are consistent for these statistics.

Findings

Limit distributions derived for degenerate U- and V-statistics

Bootstrap methods shown to be consistent for these statistics

Applications to hypothesis testing demonstrated

Abstract

We devise a general result on the consistency of model-based bootstrap methods for U- and V-statistics under easily verifiable conditions. For that purpose, we derive the limit distributions of degree-2 degenerate U- and V-statistics for weakly dependent $\mathbb{R}^d$-valued random variables first. To this end, only some moment conditions and smoothness assumptions concerning the kernel are required. Based on this result, we verify that the bootstrap counterparts of these statistics have the same limit distributions. Finally, some applications to hypothesis testing are presented.