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

TL;DR

This paper develops generalized renewal theorems for dependent interarrival times in point processes, unifying existing theories and enabling applications in fractal and hyperbolic geometry, including Minkowski measurability of self-conformal sets.

Contribution

It introduces new renewal theorems for processes with dependent interarrival times, extending classical results and connecting renewal theory with geometric applications.

Findings

Unified renewal theorems for dependent interarrival times

Application to Minkowski measurability of self-conformal sets

Generalization of Lalley's renewal theorem

Abstract

Renewal theorems are developed for point processes with interarrival times $W_n=\xi(X_{n+1}X_n\cdots)$, where $(X_n)_{n\in\mathbb Z}$ is a stochastic process with finite state space $\Sigma$ and $\xi\colon\Sigma_A\to\mathbb R$ is a H\"older continuous function on a subset $\Sigma_A\subset\Sigma^{\mathbb N}$. 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.