Strong renewal theorem and local limit theorem in the absence of regular variation

TL;DR

This paper extends renewal and local limit theorems to distributions without regular variation, covering a broader class of semistable laws with infinite mean and providing asymptotics for the renewal function.

Contribution

It establishes strong renewal theorems and local limit theorems for semistable laws with index /2, beyond the regular variation assumption, including cases with infinite mean.

Findings

Renewal theorems hold for /2 <   with infinite mean.

Local limit theorems are valid for   , covering the entire range of  .

Asymptotics of the renewal function are derived for  .

Abstract

We obtain a strong renewal theorem with infinite mean beyond regular variation, when the underlying distribution belongs to the domain of geometric partial attraction a semistable law with index $\alpha\in (1/2,1]$. In the process we obtain local limit theorems for both finite and infinite mean, that is for the whole range $\alpha\in (0,2)$. We also derive the asymptotics of the renewal function for $\alpha\in (0,1]$.