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

Let (S-n)(ngreater than or equal to0) be a Z(d)-random walk and (xi(x))(xis an element ofZd) 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 Sigma(t,j=0)([nt])h(xi(Si), xi(Sj)), t is an element of [0, 1]. The walls steps will be essentially assumed centered and the space dimension d = 2 or greater than or equal to3. (C) 2002 Elsevier Science B.V. All rights reserved.