Empirical processes of $U$-statistic structure were introduced by Serfling and studied in detail by Silverman, who proved weak convergence of weighted versions in the i.i.d. case. Our main theorem shows that this result can be generalized in two directions: First, the i.i.d. assumption can be omitted, and second, our proofs holds for a richer class of weight functions. In addition, we obtain almost sure convergence of weighted $U$-processes in the i.i.d. case which improves the results of Helmers, Janssen and Serfling, Aerts, Janssen and Mason and (in the special situation of the real line) Nolan and Pollard.
"Weak Convergence and Glivenko-Cantelli Results for Weighted Empirical $U$-Processes." Ann. Probab. 21 (2) 1170 - 1184, April, 1993. https://doi.org/10.1214/aop/1176989287