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

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