Weak convergence of a sequence of stochastic processes related with U- statistics (Q915257)

We study the weak convergence of the following sequence: \[ U_ n(t)=\left( \begin{matrix} n\\ m\end{matrix} \right)^{-1}\sum_{C_ m^{[nt]}}h(X_{i_ 1},...,X_{i_ m}) \] where \(X_ 1,...,X_ n\) is an i.i.d. sequence and [ ] is the greatest integer function. We compare \textit{A. Mandelbaum} and \textit{M. S. Taqqu}'s results in Ann. Stat. 12, 483-496 (1984; Zbl 0547.60039) with ours. The martingale method used in the proof can be extended to study infinite order U-statistics introduced by \textit{E. W. Frees} [Scand. J. Stat. 16, No.1, 29-45 (1989; Zbl 0673.62032)].