This paper introduces a new stochastic process, a collection of $U$-statistics indexed by a family of symmetric kernels. Conditions are found for the uniform almost-sure convergence of a sequence of such processes. Rates of convergence are obtained. An application to cross-validation in density estimation is given. The proofs adapt methods from the theory of empirical processes.
"$U$-Processes: Rates of Convergence." Ann. Statist. 15 (2) 780 - 799, June, 1987. https://doi.org/10.1214/aos/1176350374