The strong renewal theorem with infinite mean via local large deviations

TL;DR

This paper establishes a necessary and sufficient condition for the strong renewal theorem to hold in asymptotically stable renewal processes with infinite mean, covering all alpha in (0, 1), thus completing previous partial results.

Contribution

It provides a complete characterization of when the strong renewal theorem applies to processes with infinite mean for all alpha in (0, 1).

Findings

Established a necessary and sufficient condition for the strong renewal theorem.

Extended the validity of the theorem to all alpha in (0, 1).

Completes previous partial results from 1963.

Abstract

A necessary and sufficient condition is established for an asymptotically stable renewal process to satisfy the strong renewal theorem. This result is valid for all alpha in (0, 1), thus completing a result for alpha in (1/2, 1) which was proved in the 1963 paper of Garsia and Lamperti [6]. This paper is superseded by arXiv:1612.07635.