Asymptotic behavior of some strongly critical decomposable 3-type Galton-Watson processes with immigration

TL;DR

This paper investigates the long-term behavior of a specific critical 3-type Galton-Watson process with immigration, revealing new limit processes involving squared Bessel processes and their integrals.

Contribution

It introduces a novel limit distribution involving 2-fold iterated integrals of squared Bessel processes for critical decomposable 3-type Galton-Watson processes with immigration.

Findings

Weak convergence of scaled processes to limit involving squared Bessel processes

Identification of 2-fold iterated integral process as a new phenomenon

Extension of classical results by Foster and Ney (1978)

Abstract

We study the asymptotic behaviour of a critical decomposable 3-type Galton-Watson process with immigration when its offspring mean matrix is triangular with diagonal entries 1. 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 such a Galton-Watson process converges weakly. The limit process can be described using independent squared Bessel processes $(\mathcal{X}_{t,1})_{t\geq0}$, $(\mathcal{X}_{t,2})_{t\geq0}$, and $(\mathcal{X}_{t,3})_{t\geq0}$, the linear combinations of the integral processes of $(\mathcal{X}_{t,1})_{t\geq0}$ and $(\mathcal{X}_{t,2})_{t\geq0}$, and possibly the 2-fold iterated integral process of $(\mathcal{X}_{t,1})_{t\geq0}$. The presence of the 2-fold iterated integral process in the limit distribution is a new phenomenon in the description of asymptotic behavior of critical multi-type Galton-Watson processes with immigration. Our results complete and extend some results of Foster and Ney (1978) for some strongly critical decomposable 3-type Galton-Watson processes with immigration.