Explosion of Branching Processes

Abstract

We provide sufficient criteria for explosion in an age-dependent branching pro- cess. For this, we assume the offspring distribution is a variation of a Pareto distribution, as the chance to get at least k children is a slowly varying function over k. Given this form, we will construct a lower bound on the generation sizes. After we obtained this lower bound, we will start a process in which we will thin the tree. We will do this by pruning the children of an individual if that individual is born after its assigned time. Doing this for all generations gives rise to a thinned tree, with infinitely many non-empty generations in a finite time. Because of that, there are infinitely many individuals at a finite time, and thus the tree exploded. There is a constraint needed, dependent on the offspring distribution, for the thinned tree to survive. The goal of this thesis is to find that constraint.