Notes on Random Walks in the Cauchy Domain of Attraction

TL;DR

This paper investigates random walks within the Cauchy domain of attraction, providing new inequalities, large deviation results, and renewal theorems, with extensions to broader stable law domains.

Contribution

It introduces new probabilistic inequalities and large deviation theorems specific to the Cauchy domain, filling gaps in existing literature and extending results to broader stable law domains.

Findings

Established a Fuk-Nagaev inequality for the Cauchy domain

Derived a large deviation theorem for random walks in this domain

Extended renewal theorems to the case of stable laws with alpha in (0,2)

Abstract

The goal of these notes is to fill some gaps in the literature about random walks in the Cauchy domain of attraction, which has been in many cases left aside because of its additional technical difficulties. We prove here several results in that case: a Fuk-Nagaev inequality and a local version of it ; a large deviation theorem ; two types of local large deviation theorems. We also derive two important applications of these results: a sharp estimate of the tail of the first ladder epochs, and renewal theorems -- extending standard renewal theorems to the case of random walks. Most of our techniques carry through to the case of random walks in the domain of attraction of an $\alpha$-stable law with $\alpha \in(0,2)$, so we also present results in that case, since some of them seem to be missing in the literature.