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We derive a new representation for $U$- and $V$-statistics. Using thisrepresentation, the asymptotic distribution of $U$- and $V$-statistics can bederived by a direct application of the Continuous Mapping theorem. That novelapproach not only encompasses most of the results on the asymptoticdistribution known in literature, but also allows for the first time a unifyingtreatment of non-degenerate and degenerate $U$- and $V$-statistics. Moreover,it yields a new and powerful tool to derive the asymptotic distribution of verygeneral $U$- and $V$-statistics based on long-memory sequences. This will beexemplified by several astonishing examples. In particular, we shall presentexamples 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 convergenceof the empirical process. We also introduce the notion of asymptotic (non-)degeneracy which often appears in the presence of long-memory sequences.

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