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

Author

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  • Péter Kevei

    (University of Szeged)

  • Dalia Terhesiu

    (University of Leiden)

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 $$\alpha \in (1/2,1]$$ α ∈ ( 1 / 2 , 1 ] . In the process we obtain local limit theorems for both finite and infinite mean, that is, for the whole range $$\alpha \in (0,2)$$ α ∈ ( 0 , 2 ) . We also derive the asymptotics of the renewal function for $$\alpha \in (0,1]$$ α ∈ ( 0 , 1 ] .

Suggested Citation

  • Péter Kevei & Dalia Terhesiu, 2022. "Strong Renewal Theorem and Local Limit Theorem in the Absence of Regular Variation," Journal of Theoretical Probability, Springer, vol. 35(2), pages 1013-1048, June.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:2:d:10.1007_s10959-021-01081-w
    DOI: 10.1007/s10959-021-01081-w
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    References listed on IDEAS

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    1. Péter Kevei & Sándor Csörgő, 2009. "Merging of Linear Combinations to Semistable Laws," Journal of Theoretical Probability, Springer, vol. 22(3), pages 772-790, September.
    2. Péter Kevei & Dalia Terhesiu, 2020. "Darling–Kac Theorem for Renewal Shifts in the Absence of Regular Variation," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2027-2060, December.
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    1. Péter Kevei & Dalia Terhesiu, 2020. "Darling–Kac Theorem for Renewal Shifts in the Absence of Regular Variation," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2027-2060, December.

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