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 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 $\alpha\in (0,1]$.