  Let (S_n)_{n\in\N} be a Z-valued random walk with increments from the domain
of attraction of some \alpha-stable law and let (\xi(i))_{i\in\Z} be a sequence
of iid random variables. We want to investigate U-statistics indexed by the
random walk S_n, that is U_n:=\sum_{1\leq i<j\leq n}h(\xi(S_i),\xi(S_j)) for
some symmetric bivariate function h. We will prove the weak convergence without
assumption of finite variance. Additionally, under the assumption of finite
moments of order greater than two, we will establish a law of the iterated
logarithm for the U-statistic U_n.
