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Renewal theorems for processes with dependent interarrival times

Part of: Classical measure theory Special processes

Published online by Cambridge University Press:  29 November 2018

Sabrina Kombrink
Sabrina Kombrink*
Affiliation:
Universität zu Lübeck and Georg-August-Universität Göttingen
*
* Postal address: Mathematical Institute, Georg-August-Universität Göttingen, Bunsenstrasse 3-5, 37073 Göttingen, Germany. Email address: sabrina.kombrink@mathematik.uni-goettingen.de
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Abstract

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In this paper we develop renewal theorems for point processes with interarrival times ξ(Xn+1Xn…), where (Xn)n∈ℤ is a stochastic process with finite state space Σ and ξ:ΣA→ℝ is a Hölder continuous function on a subset ΣA⊂Σℕ. The theorems developed here unify and generalise the key renewal theorem for discrete measures and Lalley's renewal theorem for counting measures in symbolic dynamics. Moreover, they capture aspects of Markov renewal theory. The new renewal theorems allow for direct applications to problems in fractal and hyperbolic geometry, for instance to the problem of Minkowski measurability of self-conformal sets.

Keywords

Renewal theoremdependent interarrival timesymbolic dynamicskey renewal theoremRuelle‒Perron‒Frobenius theory

MSC classification

Primary: 60K05: Renewal theory 60K15: Markov renewal processes, semi-Markov processes
Secondary: 28A80: Fractals 28A75: Length, area, volume, other geometric measure theory

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 4 , December 2018 , pp. 1193 - 1216
Copyright
Copyright © Applied Probability Trust 2018 

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