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

Péter Kevei () and Dalia Terhesiu ()
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Péter Kevei: Bolyai Institute
Dalia Terhesiu: University of Exeter

Journal of Theoretical Probability, 2020, vol. 33, issue 4, 2027-2060

Abstract: 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.

Keywords: Darling–Kac law; Semistable law; Gibbs–Markov map; Renewal theorem; 37A50; 60F05 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10959-019-00930-z

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