Explosion and non-explosion in pure birth Crump--Mode--Jagers branching processes

TL;DR

This paper establishes a near-necessary condition for non-explosion in pure birth Crump--Mode--Jagers branching processes, clarifying when explosion occurs and providing a counterexample to previous assumptions.

Contribution

It offers an explicit sufficient condition for non-explosion and constructs a counterexample that addresses an open question in the theory.

Findings

Standard explosion condition is nearly necessary for rate sequences without excessive oscillations.

Counterexample shows the explosion condition is not necessary in general.

Provides insights into the structure of preferential attachment trees without infinite vertices.

Abstract

In this short note, we provide an explicit sufficient condition for non-explosion of Crump--Mode--Jagers branching processes with pure birth reproduction. It shows that the standard sufficient condition for explosion, namely the convergence of the series of reciprocals of the birth rates, is -- at least for rate sequences without excessive oscillations -- remarkably close to being necessary. At the same time, it is not necessary in full generality: we construct a counterexample which also yields a general preferential attachment tree without fitness with an infinite path and no vertices of infinite degree, thereby answering an open question previously raised in the literature.