  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.
