Asymptotic behaviour of critical decomposable 2-type Galton-Watson processes with immigration. (English) Zbl 1514.60096
Summary: 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 J. Foster and P. Ney [Z. Wahrscheinlichkeitstheor. Verw. Geb. 46, 13–43 (1978; Zbl 0373.60104)] for some strongly critical decomposable 2-type Galton-Watson processes with immigration.
MSC:
| 60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |
| 60F17 | Functional limit theorems; invariance principles |
Keywords:
multi-type Galton-Watson process with immigration; critical; decomposable; asymptotic behaviour; squared Bessel process; stationary distributionCitations:
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