Another characterization of homogeneous Poisson processes

TL;DR

This paper characterizes when the first renewal epochs of thinned processes are independent, showing that this independence, combined with minimal conditions, uniquely identifies homogeneous Poisson processes.

Contribution

It provides an analytic characterization of homogeneous Poisson processes based on the independence of first renewal epochs after Bernoulli thinning.

Findings

Independence of first renewal epochs characterizes homogeneous Poisson processes.

The proof is analytic in nature.

Minimal extra conditions suffice for identification.

Abstract

For a general renewal process $N$ (allowing delay, defect and multiple simultaneous arrivals) the independence of the first renewal epochs of the marked processes got from $N$ by Bernoulli $0$/$1$ thinning is characterized. This independence is well-known to hold true in the case of homogeneous Poisson processes; by way of corollary one obtains the interesting observation that, when coupled with some minimal extra conditions, it in fact already identifies them. The proof is analytic in character.