Finitary isomorphisms of renewal point processes and continuous-time regenerative processes

TL;DR

This paper demonstrates that many stationary continuous-time regenerative processes, including certain renewal point processes, are finitarily isomorphic to Poisson processes, providing new conditions and improving previous results.

Contribution

It establishes conditions under which renewal point processes are finitarily isomorphic to Poisson processes, advancing understanding of their structural similarities.

Findings

Renewal point processes with exponential tail jump distributions are finitarily isomorphic to Poisson processes.

Provides necessary and sufficient conditions for finitary isomorphism to Poisson processes.

Answers open questions from prior research by Soo and Kosloff and Soo.

Abstract

We show that a large class of stationary continuous-time regenerative processes are finitarily isomorphic to one another. The key is showing that any stationary renewal point process whose jump distribution is absolutely continuous with exponential tails is finitarily isomorphic to a Poisson point process. We further give simple necessary and sufficient conditions for a renewal point process to be finitarily isomorphic to a Poisson point process. This improves results and answers several questions of Soo and of Kosloff and Soo.