Local large deviations and the strong renewal theorem

Francesco Caravenna, Ron Doney

Published 2016-12-22Version 1

We establish two different, but related results for random walks in the domain of attraction of a stable law of index $\alpha$. The first result is a local large deviation upper bound, valid for $\alpha \in (0,1) \cup (1,2)$, which improves on the classical Gnedenko and Stone local limit theorems. The second result, valid for $\alpha \in (0,1)$, is the derivation of necessary and sufficient conditions for the random walk to satisfy the strong renewal theorem (SRT). This solves a long standing problem, which dates back to the 1963 paper of Garsia and Lamperti \cite{cf:GL} for renewal processes (i.e. random walks with non-negative increments), and to the 1968 paper of Williamson \cite{cf:Wil} for general random walks. This paper supersedes the individual preprints arXiv:1507.07502 and arXiv:1507.06790