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 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.

Renewal function asymptotics refined à la Feller

RENEWAL FUNCTION ASYMPTOTICS REFINED À LA FELLERFeller’s volume 2 shows how to use the Key Renewal Theorem to prove that in the limit x!1, the renewal function Ux of a renewal process with nonarithmetic generic lifetime X with finite mean EX=1=and second moment differs from its linear asymptote x by the quantity 122EX2. His first edition 1966 but not the second in 1971 asserted that a similar approach would refine this asymptotic result when X has finite higher order moments. The paper shows how higher order moments may justify drawing conclusions from a recurrence relation that exploits a general renewal equation and further appeal to the Key Renewal Theorem.

A Diffusion Approximation for Markov Renewal Processes

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.

A Diffusion Approximation for Markov Renewal Processes

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.

A Diffusion Approximation for Markov Renewal Processes

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.