A complete coupling proof of Blackwell's renewal theorem

Abstract

Blackwell's renewal theorem for non-lattice renewal processes with mean tecurrence time m states the expected number of renewals in a time-interval of length h tends to h m as the interval goes to infinity: E[N(t,t+;h]]→ h m , t→∞ This note presents a self-contained coupling proof of this result mending the drawbacks of earlier such proofs. Firstly, the proof is complete in the sense that it covers not only the case m<∞ but also m=∞. Secondly, the proof is fairly elementary in the sense that it does not rely on advanced results such as Hewitt-Savage 0–1-Law or the ϵ-recurrence of 0-mean random walks.