  We establish limit theorems for U-statistics indexed by a random walk on Z^d
and we express the limit in terms of some Levy sheet Z(s,t). Under some
hypotheses, we prove that the limit process is Z(t,t) if the random walk is
transient or null-recurrent ant that it is some stochastic integral with
respect to Z when the walk is positive recurrent. We compare our results with
results for random walks in random scenery.
