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

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Acknowledgements

We are thankful to Vilmos Totik for showing us a simpler proof of the strict positivity of the real part in Theorem 1 and to the anonymous referee for the remarks and suggestions, in particular for pointing out reference [22].

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Correspondence to Péter Kevei.

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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

Keywords

  • Local limit theorem
  • Strong renewal theorem
  • Semistable law

Mathematics Subject Classification (2020)

  • 60K05