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

TL;DR

This paper investigates the asymptotic behavior of U-statistics indexed by a heavy-tailed random walk, proving weak convergence without finite variance and establishing a law of the iterated logarithm under finite moments.

Contribution

It introduces a novel analysis of U-statistics indexed by a random walk with stable domain of attraction, extending classical results to heavy-tailed settings.

Findings

Proves weak convergence of U-statistics without finite variance assumptions.

Establishes a law of the iterated logarithm for U-statistics with finite moments.

Extends limit theorems to processes indexed by heavy-tailed random walks.

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.