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 \(\alpha \in (0,1]\).
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PK’s research was partially supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, by the NKFIH Grant FK124141, and by the EU-funded Hungarian Grant EFOP-3.6.1-16-2016-00008. The research of DT was partially supported by EPSRC Grant EP/S019286/1.
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Kevei, P., Terhesiu, D. Strong Renewal Theorem and Local Limit Theorem in the Absence of Regular Variation. J Theor Probab 35, 1013–1048 (2022). https://doi.org/10.1007/s10959-021-01081-w
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DOI: https://doi.org/10.1007/s10959-021-01081-w