Wichura (1969) studied an invariance principle for partial sums of a multi-dimensional array of independent random variables. It is shown that a similar invariance principle holds for a broad class of generalized $U$-statistics for which the different terms in the partial sums are not independent. Weak convergence of generalized $U$-statistics for random sample sizes is also studied. The case of (generalized) von Mises' functional is treated briefly.
"Weak Convergence of Generalized $U$-Statistics." Ann. Probab. 2 (1) 90 - 102, February, 1974. https://doi.org/10.1214/aop/1176996754