The strong renewal theorem

TL;DR

This paper provides a complete characterization of when the strong renewal theorem holds for positive-increment random walks in the domain of attraction of a stable law with index less than one, resolving a long-standing open problem.

Contribution

It offers explicit necessary and sufficient conditions for the strong renewal theorem to hold when the stability index is at most 1/2, including cases with negative increments.

Findings

Conditions fail for negative-valued random walks.

Complete solution for positive increments case.

Independent proof by Doney confirms results.

Abstract

We consider real random walks with positive increments (renewal processes) in the domain of attraction of a stable law with index $\alpha \in (0,1)$. The famous local renewal theorem of Garsia and Lamperti, also called strong renewal theorem, is known to hold in complete generality only for $\alpha > \frac{1}{2}$. Understanding when the strong renewal theorem holds for $\alpha \le \frac{1}{2}$ is a long-standing problem, with sufficient conditions given by Williamson, Doney and Chi. In this paper we give a complete solution, providing explicit necessary and sufficient conditions (an analogous result has been independently and simultaneously proved by Doney in arXiv:1507.06790). We also show that these conditions fail to be sufficient if the random walk is allowed to take negative values. This paper is superseded by arXiv:1612.07635