Convergence of U-statistics indexed by a random walk to stochastic integrals of a Levy sheet

A U-statistic indexed by a Z(d0)-random walk (S-n)(n) is a process U-n := Sigma(n)(i,j=1) h(xi S-i, xi S) where his some real-valued function and (xi(k))(k) is a sequence of i.i.d. random variables, which are independent of the walk. Concerning the walk, we assume either that it is transient or that its increments are in the normal domain of attraction of a strictly stable distribution of exponent alpha is an element of [d(0), 2]. We further assume that the distribution of h(xi(1), xi(2)) belongs to the normal domain of attraction of a strictly stable distribution of exponent beta is an element of (0, 2). For a suitable renormalization (a(n))(n) we establish the convergence in distribution of the sequence of processes (Uleft perpendicularntright perpendicular/a(n))t; n is an element of N to some suitable observable of a Levy sheet (Z(s,t))(s,t). The limit process is the diagonal process (Z(t,t))(t) when alpha = d(0) is an element of {1, 2} or when the underlying walk is transient for arbitrary d(0) >= 1. When alpha >= d(0) = 1, the limit process is some stochastic integral with respect to Z.