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

Abstract We study the asymptotic behavior of a critical decomposable 3-type Galton–Watson process with immigration when its offspring mean matrix is triangular with diagonal entries all 1. It is proved that, under second-order 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 with the use of independent squared Bessel processes left parenthesis script upper X Subscript t comma 1 Baseline right parenthesis Subscript t greater than or equals 0 ( X t , 1 ) t ≥ 0 $({\mathcal X}_{t,1})_{t\geq0}$ , left parenthesis script upper X Subscript t comma 2 Baseline right parenthesis Subscript t greater than or equals 0 ( X t , 2 ) t ≥ 0 $({\mathcal X}_{t,2})_{t\geq0}$ , and left parenthesis script upper X Subscript t comma 3 Baseline right parenthesis Subscript t greater than or equals 0 ( X t , 3 ) t ≥ 0 $({\mathcal X}_{t,3})_{t\geq0}$ , the linear combinations of the integral processes of left parenthesis script upper X Subscript t comma 1 Baseline right parenthesis Subscript t greater than or equals 0 ( X t , 1 ) t ≥ 0 $({\mathcal X}_{t,1})_{t\geq0}$ and left parenthesis script upper X Subscript t comma 2 Baseline right parenthesis Subscript t greater than or equals 0 (