Abstract
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 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.
Original language | English |
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Pages (from-to) | 1193-1216 |
Number of pages | 23 |
Journal | Advances in Applied Probability |
Volume | 50 |
Issue number | 4 |
Early online date | 29 Nov 2018 |
DOIs | |
Publication status | Published - 1 Dec 2018 |
Keywords
- Renewal theory
- dependent interarrival time
- symbolic dynamics
- Ruelle–Perron–Frobenius theory
ASJC Scopus subject areas
- General Mathematics