Renewal theorems for processes with dependent interarrival times

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...

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Bibliographic Details
Published inAdvances in applied probability Vol. 50; no. 4; pp. 1193 - 1216
Main Author Kombrink, Sabrina
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.12.2018
Applied Probability Trust
Subjects
Continuity (mathematics)
Fractal geometry
Markov analysis
Markov processes
Original Article
Probability
Set theory
Stochastic processes
Theorems
dependent interarrival time
Secondary 28A80
28A75
Renewal theorem
Primary 60K05
key renewal theorem
60K15
symbolic dynamics
Ruelle‒Perron‒Frobenius theory
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Summary: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.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
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content type line 14
ISSN:0001-8678
1475-6064
DOI:10.1017/apr.2018.56