Notes on Random Walks in the Cauchy Domain of Attraction

Quentin Berger (1)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

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 many of them seem to be missing in the literature.

Mots clés: Ladder Epochs; Renewal Theorems; Local Large Deviations; Fuk-Nagaev Inequalities; Stable Distribution; Cauchy Domain of Attraction; Random Walks

Domaines: Probabilités [math.PR]; Physique mathématique [math-ph]

Quentin Berger. Notes on Random Walks in the Cauchy Domain of Attraction. Probability Theory and Related Fields, 2019, 175 (1-2), pp.1-42. ⟨10.1007/s00440-018-0887-0⟩. ⟨hal-01576409v3⟩