Notes on random walks in the Cauchy domain of attraction

The goal of these notes is to fill some gaps in the literature about random walks in the Cauchy domain of attraction, which has often been 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. Most of our techniques carry through to the case of random walks in the domain of attraction of an alpha-stable law with alpha is an element of (0, 2), so we also present results in that case-some of them are improvement of the existing literature.