Recurrence properties of a special type of heavy-tailed random walk

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. How- ever, 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 appropri- ate 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.