Stable limit theorem for $U$-statistic processes indexed by a random walk

Abstract

Let $(S_n)_{n\in \mathbb{N} }$ be a $\mathbb{Z} $-valued random walk with increments from the domain of attraction of some $\alpha $-stable law and let $(\xi (i))_{i\in \mathbb{Z} }$ 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 $U_n$.