Multitype branching process with nonhomogeneous Poisson and generalized Polya immigration

TL;DR

This paper studies multitype branching processes with immigrants arriving via nonhomogeneous Poisson or generalized Polya processes, demonstrating convergence of normalized populations and analyzing moments in two-type cases.

Contribution

It introduces a framework for multitype branching processes with complex immigration modeled by nonhomogeneous Poisson and Polya processes, proving convergence results and transient moment analysis.

Findings

Normalized populations converge in distribution for supercritical, critical, and subcritical cases.

Provides transient moment analysis for two-type particle systems.

Establishes a unified approach for different immigration processes in branching models.

Abstract

In a multitype branching process, it is assumed that immigrants arrive according to a nonhomogeneous Poisson or a generalized Polya process (both processes are formulated as a nonhomogeneous birth process with an appropriate choice of transition intensities). We show that the renormalized numbers of objects of the various types alive at time $t$ for supercritical, critical, and subcritical cases jointly converge in distribution under those two different arrival processes. Furthermore, some transient moment analysis when there are only two types of particles is provided. AMS 2000 subject classifications: Primary 60J80, 60J85; secondary 60K10, 60K25, 90B15.