Strong renewal theorems with infinite mean beyond local large deviations

TL;DR

This paper proves the strong renewal theorem for certain heavy-tailed distributions with infinite mean, extending large deviations techniques and weakening previous boundedness conditions, with applications to ladder processes and infinitely divisible distributions.

Contribution

It extends the strong renewal theorem to distributions in the domain of attraction of a stable law with exponent lpha  (0, 1/2], using weaker conditions than prior work.

Findings

Established the strong renewal theorem under weaker conditions.

Extended large deviations approach to broader distribution classes.

Applied results to ladder height processes and infinitely divisible distributions.

Abstract

Let $F$ be a distribution function on the line in the domain of attraction of a stable law with exponent $\alpha\in(0,1/2]$. We establish the strong renewal theorem for a random walk $S_1,S_2,\ldots$ with step distribution $F$, by extending the large deviations approach in Doney [Probab. Theory Related Fileds 107 (1997) 451-465]. This is done by introducing conditions on $F$ that in general rule out local large deviations bounds of the type $\mathbb{P}\{S_n\in(x,x+h]\}=O(n)\overline{F}(x)/x$, hence are significantly weaker than the boundedness condition in Doney (1997). We also give applications of the results on ladder height processes and infinitely divisible distributions.