Weak Convergence of Generalized U-Statistics

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.