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

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