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

Let (Sn)n⩾0 be a Zd-random walk and (ξx)x∈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 R2 with values in R. We study the weak convergence of the sequence Un, n∈N, with values in D[0,1] the set of right continuous real-valued functions with left limits, defined by∑i,j=0[nt] h(ξSi,ξSj), t∈[0,1].The walk steps will be essentially assumed centered and the space dimension d=2 or ⩾3.