Asymptotic behavior of critical primitive multi-type branching processes with immigration

TL;DR

This paper proves that under certain conditions, critical multi-type branching processes with immigration can be approximated by a squared Bessel process, revealing their asymptotic behavior through diffusion limits.

Contribution

It establishes a weak convergence result for scaled multi-type branching processes with immigration to a squared Bessel process, under the assumption of a primitive offspring mean matrix.

Findings

Convergence of scaled processes to squared Bessel process

Identification of the limiting process as a diffusion on a ray

Extension of diffusion approximation to multi-type 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 towards a squared Bessel process supported by a ray determined by the Perron vector of the offspring mean matrix.