Limit theorems for random symmetric functions

In this paper, based on theorems for limit distributions of empirical power processes for the i.i.d. case and for the case with independent triangular arrays of random variables, we prove limit theorems for U- and V-statistics determined by generalized polynomial kernel functions. We also show that under some natural conditions the limit distributions can be represented as functionals on the limit process of the normed empirical power process. We consider the one-sample case, as well as multi-sample cases. Dedicated to Professor V. M. Zolotarev on his sixty-fifth birthday. Supported by the Hungarian National Foundation for Scientific Research (grant No. T1666). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part I.