Asymptotic Behavior of Critical Primitive Multi-Type Branching Processes with Immigration

Abstract

Under natural assumptions a Feller-type diffusion approximation is derived for critical multi-type branching processes with immigration when the offspring mean matrix is primitive (in other words, positively regular). Namely, it is proved that a sequence of appropriately scaled random step functions formed from a sequence of critical primitive multi-type branching processes with immigration converges weakly toward a squared Bessel process supported by a ray determined by the Perron vector of the offspring mean matrix.

Keywords:

• Critical primitive multi-type branching processes with immigration
• Squared Bessel processes

2000 Mathematics Subject Classifications:

• Primary: 60J80
• 60F17
• Secondary: 60J60

Additional information

Funding

M. Ispány has been supported by the TÁMOP-4.2.2.C-11/1/KONV-2012-0001 project. The project has been supported by the European Union, co-financed by the European Social Fund. The research of G. Pap was realized in the frames of TÁMOP 4.2.4. A/2-11-1-2012-0001, “National Excellence Program–Elaborating and operating an inland student and researcher personal support system.” The project was subsidized by the European Union and co-financed by the European Social Fund.