Decomposition of supercritical linear-fractional branching processes

TL;DR

This paper analyzes the decomposition of supercritical linear-fractional Bienyamé-Galton-Watson processes with countably many types, extending known results from single-type cases to more complex multi-type scenarios.

Contribution

It introduces a decomposition framework for supercritical linear-fractional multi-type branching processes, generalizing the single-type decomposition to countably many types.

Findings

Decomposition into infinite and finite descent subtypes for multi-type processes

Extension of known single-type results to countably many types

Framework for analyzing supercritical linear-fractional processes

Abstract

It is well known that a supercritical single-type Bienyam\'e-Galton-Watson process can be viewed as a decomposable branching process formed by two subtypes of particles: those having infinite line of descent and those who have finite number of descendants. In this paper we analyze such a decomposition for the linear-fractional Bienyam\'e-Galton-Watson processes with countably many types.