Strong Renewal Theorem and Local Limit Theorem in the Absence of Regular Variation

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 of 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 \(\alpha \in (0,1]\).