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

Let (Sn)n[greater-or-equal, slanted]0 be a -random walk and be a sequence of independent and identically distributed -valued random variables, independent of the random walk. Let h be a measurable, symmetric function defined on with values in . We study the weak convergence of the sequence , with values in D[0,1] the set of right continuous real-valued functions with left limits, defined byThe walk steps will be essentially assumed centered and the space dimension d=2 or [greater-or-equal, slanted]3.