Diffusion approximation of critical controlled multi-type branching processes

TL;DR

This paper derives a diffusion approximation for critical controlled multi-type branching processes, showing convergence to a squared Bessel process and analyzing the asymptotic behavior of type frequencies.

Contribution

It introduces a Feller-type diffusion approximation for critical CMBPs under linearity assumptions, extending to processes with immigration and two-sex models.

Findings

Weak convergence to squared Bessel process

Diffusion approximation for multi-type processes with immigration

Asymptotic behavior of type frequencies

Abstract

Branching processes form an important family of stochastic processes that have been successfully applied in many fields. In this paper, we focus our attention on controlled multi-type branching processes (CMBPs). A Feller-type diffusion approximation is derived for some critical CMBPs. Namely, we consider a sequence of appropriately scaled random step functions formed from a critical CMBP with control distributions having expectations that satisfy a kind of linearity assumption. It is proved that such a sequence converges weakly toward a squared Bessel process supported by a ray determined by an eigenvector of a matrix related to the offspring mean matrix and the control distributions of the branching process in question. As applications, among others, we derive Feller-type diffusion approximations of critical, primitive multi-type branching processes with immigration and some two-sex branching processes. We also describe the asymptotic behaviour of the relative frequencies of distinct types of individuals for critical CMBPs.