Explosion and non-explosion in pure birth Crump--Mode--Jagers branching processes
Oleksii Galganov, Andrii Ilienko
TL;DR
The paper analyzes explosion and non-explosion in pure birth Crump--Mode--Jagers processes and links to fitness-free preferential attachment trees. It derives explicit explosion and non-explosion criteria based on the rate sequence $(\lambda_i)$, and constructs a pathological rate sequence showing that explosion can occur even when the reciprocal-sum diverges, while not exhibiting infinite stars. It shows the standard explosion criterion $\sum_i\lambda_i^{-1}<\infty$ is nearly necessary in non-oscillatory regimes and uses this to characterize possible tree shapes without fitness, including a unique infinite path and no infinite stars. These results address an open question about the shapes of preferential attachment trees without fitness and clarify the boundary between explosive and non-explosive CMJ processes.
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.