Limit distributions of V- and U-statistics in terms of multiple stochastic Wiener-type integrals

Abstract

We describe the limit distribution of V- and U-statistics in a new fashion. In the case of V-statistics the limit variable is a multiple stochastic integral with respect to an abstract Brownian bridge G Q . This extends the pioneer work of Filippova (1961) [8]. In the case of U-statistics we obtain a linear combination of G Q -integrals with coefficients stemming from Hermite Polynomials. This is an alternative representation of the limit distribution as given by Dynkin and Mandelbaum (1983) [7] or Rubin and Vitale (1980) [13]. It is in total accordance with their results for product kernels.