A special family of Galton-Watson processes with explosions

TL;DR

This paper introduces a four-parameter extension of linear-fractional Galton-Watson processes, enabling explicit analysis of processes with potential infinite mean or offspring, and explores their explosive behavior and explosion timing distribution.

Contribution

It extends the classical two-parameter family to a richer four-parameter family, allowing explicit calculations for processes with infinite mean or offspring, and analyzes explosion times.

Findings

Explicit formulas for the extended family of processes.

Identification of conditions leading to explosion.

Approximation of explosion time distribution by Gumbel distribution.

Abstract

The linear-fractional Galton-Watson processes is a well known case when many characteristics of a branching process can be computed explicitly. In this paper we extend the two-parameter linear-fractional family to a much richer four-parameter family of reproduction laws. The corresponding Galton-Watson processes also allow for explicit calculations, now with possibility for infinite mean, or even infinite number of offspring. We study the properties of this special family of branching processes, and show, in particular, that in some explosive cases the time to explosion can be approximated by the Gumbel distribution.