A multi-type branching process with immigration and random environment

Abstract

A population of several types of particles is considered where immigration of the particles into the population is allowed. This immigration causes “fragmentation” or branching of all the existing target particles in the population. An immigrating particle can also fragment into the different types. We will obtain results concerning the random vector N m =[ N(1, m),…, N( k, m)] whose r-th component is a random variable equal to the number of particles of type r in the m-th generation, where mϵ M={0,1,2,…} . For the case where the particles develop in discrete random environment (R.E.), or random environment with immigration of the various types (I.R.E.), conditions for the almost sure convergence of N mþ −m , as m ar…, are given where ρ > 1 is the largest eigenvalue of a first moment matrix Λ. We make special note of the fact that this complements the author's previous work [2,3], for R.E. processes with ρ⩽1, and that this work also extends that of Kesten and Stigum [5] to provide necessary and sufficient conditions for the almost sure convergence of the ordinary k-type process with immigration.