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.
Deborah Nolan. David Pollard. "Functional Limit Theorems for $U$-Processes." Ann. Probab. 16 (3) 1291 - 1298, July, 1988. https://doi.org/10.1214/aop/1176991691