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

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.