Local large deviations and the strong renewal theorem

TL;DR

This paper advances the understanding of random walks in the domain of attraction of stable laws by establishing new local large deviation bounds and characterizing conditions for the strong renewal theorem, solving longstanding open problems.

Contribution

It provides the first local large deviation upper bounds for a broad class of stable law indices and characterizes necessary and sufficient conditions for the strong renewal theorem for these walks.

Findings

Improved local limit theorems for stable law domains

Necessary and sufficient conditions for the strong renewal theorem

Resolution of a long-standing problem in renewal theory

Abstract

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 1962 paper of Garsia and Lamperti [Comm. Math. Helv.] for renewal processes (i.e. random walks with non-negative increments), and to the 1968 paper of Williamson [Pacific J. Math.] for general random walks. This paper supersedes the individual preprints arXiv:1507.07502 and arXiv:1507.06790