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Darling–Kac Theorem for Renewal Shifts in the Absence of Regular Variation

Author

Listed:
  • Péter Kevei

    (Bolyai Institute)

  • Dalia Terhesiu

    (University of Exeter)

Abstract

Regular variation is an essential condition for the existence of a Darling–Kac law. We weaken this condition assuming that the renewal distribution belongs to the domain of geometric partial attraction of a semistable law. In the simple setting of one-sided null recurrent renewal chains, we derive a Darling–Kac limit theorem along subsequences. Also in this context, we determine the asymptotic behaviour of the renewal function and obtain a Karamata theorem for positive operators. We provide several examples of dynamical systems to which these results apply.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:4:d:10.1007_s10959-019-00930-z
    DOI: 10.1007/s10959-019-00930-z
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    References listed on IDEAS

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    1. István Berkes & László Györfi & Péter Kevei, 2017. "Tail Probabilities of St. Petersburg Sums, Trimmed Sums, and Their Limit," Journal of Theoretical Probability, Springer, vol. 30(3), pages 1104-1129, September.
    2. Péter Kevei & Sándor Csörgő, 2009. "Merging of Linear Combinations to Semistable Laws," Journal of Theoretical Probability, Springer, vol. 22(3), pages 772-790, September.
    3. Allan Gut & Anders Martin-Löf, 2016. "A Maxtrimmed St. Petersburg Game," Journal of Theoretical Probability, Springer, vol. 29(1), pages 277-291, March.
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    Cited by:

    1. 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.

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    1. István Berkes & László Györfi & Péter Kevei, 2017. "Tail Probabilities of St. Petersburg Sums, Trimmed Sums, and Their Limit," Journal of Theoretical Probability, Springer, vol. 30(3), pages 1104-1129, September.
    2. 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.

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