Asymptotic behaviour of critical decomposable 2-type Galton-Watson processes with immigration

TL;DR

This paper analyzes the long-term behavior of a critical decomposable 2-type Galton-Watson process with immigration, showing convergence to processes involving Bessel processes and stationary distributions under certain moment conditions.

Contribution

It extends previous results by characterizing the asymptotic behavior of decomposable 2-type Galton-Watson processes with immigration, including convergence to Bessel processes.

Findings

Weak convergence of scaled processes to Bessel processes

Identification of stationary distributions for subcritical components

Extension of Foster and Ney's results to decomposable cases

Abstract

In this paper the asymptotic behaviour of a critical 2-type Galton-Watson process with immigration is described when its offspring mean matrix is reducible, in other words, when the process is decomposable. It is proved that, under second or fourth order moment assumptions on the offspring and immigration distributions, a sequence of appropriately scaled random step processes formed from a critical decomposable 2-type Galton-Watson process with immigration converges weakly. The limit process can be described using one or two independent squared Bessel processes and possibly the unique stationary distribution of an appropriate single-type subcritical Galton-Watson process with immigration. Our results complete and extend the results of Foster and Ney (1978) for some strongly critical decomposable 2-type Galton-Watson processes with immigration.