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It is commonly acknowledged that V-functionals with an unbounded kernel arenot Hadamard differentiable and that therefore the asymptotic distribution ofU- and V-statistics with an unbounded kernel cannot be derived by theFunctional Delta Method (FDM). However, in this article we show thatV-functionals are quasi-Hadamard differentiable and that therefore a modifiedversion of the FDM (introduced recently in (J. Multivariate Anal. 101 (2010)2452--2463)) can be applied to this problem. The modified FDM requires weakconvergence of a weighted version of the underlying empirical process. Thelatter is not problematic since there exist several results on weightedempirical processes in the literature; see, for example, (J. Econometrics 130(2006) 307--335, Ann. Probab. 24 (1996) 2098--2127, Empirical Processes withApplications to Statistics (1986) Wiley, Statist. Sinica 18 (2008) 313--333).The modified FDM approach has the advantage that it is very flexible w.r.t.both the underlying data and the estimator of the unknown distributionfunction. Both will be demonstrated by various examples. In particular, we willshow that our FDM approach covers mainly all the results known in literaturefor the asymptotic distribution of U- and V-statistics based on dependent data-- and our assumptions are by tendency even weaker. Moreover, using our FDMapproach we extend these results to dependence concepts that are not covered bythe existing literature.

Deriving the asymptotic distribution of U- and V-statistics of dependent data using weighted empirical processes