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arXiv · 1512.08351

Renewal theorems for a class of processes with dependent interarrival times and applications in geometry

Sabrina Kombrink
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Abstract

Renewal theorems are developed for point processes with interarrival times $W_n=ξ(X_{n+1}X_n\cdots)$, where $(X_n)_{n\in\mathbb Z}$ is a stochastic process with finite state space $Σ$ and $ξ\colonΣ_A\to\mathbb R$ is a Hölder continuous function on a subset $Σ_A\subsetΣ^{\mathbb N}$. 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, results on the Minkowski measurability of self-conformal sets are deduced. Indeed, these geometric problems motivated the development of the renewal theorems.

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Sabrina Kombrink. 2017-05-11. Renewal theorems for a class of processes with dependent interarrival times and applications in geometry. https://doi.org/10.1017/apr.2018.56

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