Implosion of a pure death process

TL;DR

This paper investigates the behavior of a pure death process, demonstrating conditions under which extinction occurs sequentially or via a finite-time implosion, with implications for understanding such stochastic processes.

Contribution

It introduces the concept of process implosion and provides examples of extinction paths in a pure death process, including finite-time collapse from infinity.

Findings

Extinction can occur along a path where only one individual dies at a time.

A process can go from infinity to zero in finite time, demonstrating implosion.

Examples illustrate different extinction behaviors in pure death processes.

Abstract

We study a pure death process. At each discrete time every individual dies or not independently of each other with a constant probability. We give examples showing that in a certain limit extinction happens along a path where one and only one individual is lost at a time. We also exhibit an example for which such a path goes from infinity to 0 in a finite time. This is what we call a process implosion.