Multitype branching process with nonhomogeneous Poisson and contagious Poisson immigration

In a multitype branching process, it is assumed that immigrants arrive according to a nonhomogeneous Poisson or a contagious Poisson process (both processes are formulated as a nonhomogeneous birth process with an appropriate choice of transition intensities). We show that the normalized numbers of objects of the various types alive at time t for supercritical, critical, and subcritical cases jointly converge in distribution under those two dierent arrival processes. Furthermore, some transient expectation results when there are only two types of particles are provided.