For partial cumulative sums of independent and identically distributed random variables (i.i.d.r.v.) with a finite (positive) variance, weak convergence to Brownian motion processes has been established by Donsker (1951, 1952). The result is extended here to differentiable statistical functions of von Mises (1947) and $U$-statistics of Hoeffding (1948). Along with the extension to generalized $U$-statistics, a few applications are briefly sketched.
R. G. Miller Jr.. Pranab Kumar Sen. "Weak Convergence of $U$-Statistics and Von Mises' Differentiable Statistical Functions." Ann. Math. Statist. 43 (1) 31 - 41, February, 1972. https://doi.org/10.1214/aoms/1177692698