General branching processes in discrete time as random trees

TL;DR

This paper extends the classical Galton-Watson process to more complex branching processes with overlapping generations, using a tree-based perspective to analyze growth conditions and relationships between different models.

Contribution

It introduces a new tree-based framework for understanding general branching processes with overlapping generations, providing clearer analysis of growth conditions and model relations.

Findings

Size-biased description of general branching processes

Analysis of the $x\log x$ growth condition

Relation between simple and complex branching processes

Abstract

The simple Galton--Watson process describes populations where individuals live one season and are then replaced by a random number of children. It can also be viewed as a way of generating random trees, each vertex being an individual of the family tree. This viewpoint has led to new insights and a revival of classical theory. We show how a similar reinterpretation can shed new light on the more interesting forms of branching processes that allow repeated bearings and, thus, overlapping generations. In particular, we use the stable pedigree law to give a transparent description of a size-biased version of general branching processes in discrete time. This allows us to analyze the $x\log x$ condition for exponential growth of supercritical general processes as well as relation between simple Galton--Watson and more general branching processes.