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

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 $ℝ^d$-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.