Functional Limit Theorems for $U$-Processes

Abstract

A $U$-process is a collection of $U$-statistics indexed by a family of symmetric kernels. In this paper, two functional limit theorems are obtained for sequences of standardized $U$-processes. In one case the limit process is Gaussian; in the other, the limit process has finite dimensional distributions of infinite weighted sums of $\chi^2$ random variables. Goodness-of-fit statistics provide examples.