Renewal theorems for processes with dependent interarrival times

In this paper we develop renewal theorems for point processes with interarrival times ξ(X_n+1X_n···), where (X_n)_n∈Z is a stochastic process with finite state space and ξ: Σ_A → ℝ is a Hölder continuous function on a subset Σ_A ⊂ Σ^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 to the problem of Minkowski measurability of self-conformal sets.