Functional limit theorems for U-statistics indexed by a random walk

Let (Sn)(n greater than or equal to0) be a Z(d)-random walk and (xi (x))(x is an element of Zd) be a sequence of independent and identically distributed R-valued random variables, independent of the random walk. Let h be a measurable, symmetric function defined on R-2 with values in R. We study the weak convergence of the sequence U-n, n is an element of N, with values in D[0, 1], the set of right continuous real-valued functions with left limits, defined by [GRAPHICS] (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.