Recurrence properties of a special type of Heavy-Tailed Random Walk

TL;DR

This paper investigates the recurrence properties of a special heavy-tailed random walk related to Lorentz processes, providing new results for the infinite horizon case and estimating local limit theorem remainders.

Contribution

It establishes recurrence properties for a non-normal domain of attraction heavy-tailed random walk, extending known results from simpler models.

Findings

Recurrence properties are confirmed for the heavy-tailed random walk.

An estimation of the local limit theorem remainder term is provided.

Results extend understanding of Lorentz process perturbations.

Abstract

In the proof of the invariance principle for locally perturbed periodic Lorentz process with finite horizon, a lot of delicate results were needed concerning the recurrence properties of its unperturbed version. These were analogous to the similar properties of Simple Symmetric Random Walk. However, in the case of Lorentz process with infinite horizon, the analogous results for the corresponding random walk are not known, either. In this paper, these properties are ascertained for the appropriate random walk (this happens to be in the non normal domain of attraction of the normal law). As a tool, an estimation of the remainder term in the local limit theorem for the corresponding random walk is computed.