A martingale view of Blackwell's renewal theorem and its extensions to a general counting process

TL;DR

This paper employs martingale techniques to analyze counting processes, offering new representations and extending Blackwell's renewal theorem to more general processes with conditions for similar asymptotic behavior.

Contribution

It introduces a novel semimartingale representation for renewal processes and extends Blackwell's renewal theorem to general counting processes.

Findings

New semimartingale representation for renewal processes

Extended Blackwell's renewal theorem to general counting processes

Provided conditions for asymptotic behavior similar to classical renewal results

Abstract

Martingales constitute a basic tool in stochastic analysis; this paper considers their application to counting processes. We use this tool to revisit a renewal theorem and its extensions for various counting processes. We first consider a renewal process as a pilot example, deriving a new semimartingale representation that differs from the standard decomposition via the stochastic intensity function. We then revisit Blackwell's renewal theorem, its refinements and extensions. Based on these observations, we extend the semimartingale representation to a general counting process, and give conditions under which asymptotic behaviour similar to Blackwell's renewal theorem holds.