Continuous mapping approach to the asymptotics of $U$- and $V$-statistics

Abstract

We derive a new representation for $U$- and $V$-statistics. Using this representation, the asymptotic distribution of $U$- and $V$-statistics can be derived by a direct application of the Continuous Mapping theorem. That novel approach not only encompasses most of the results on the asymptotic distribution known in literature, but also allows for the first time a unifying treatment of non-degenerate and degenerate $U$- and $V$-statistics. Moreover, it yields a new and powerful tool to derive the asymptotic distribution of very general $U$- and $V$-statistics based on long-memory sequences. This will be exemplified by several astonishing examples. In particular, we shall present examples where weak convergence of $U$- or $V$-statistics occurs at the rate $a_{n}^{3}$ and $a_{n}^{4}$, respectively, when $a_{n}$ is the rate of weak convergence of the empirical process. We also introduce the notion of asymptotic (non-) degeneracy which often appears in the presence of long-memory sequences.