Stable Limit Theorem for U-Statistic Processes Indexed by a Random Walk

Abstract

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.