A note on multitype branching processes with immigration in a random environment

We consider a multitype branching process with immigration in a ran-dom environment introduced by Key in [Ann. Probab. 15 (1987) 344–353]. It was shown by Key that, under the assumptions made in [Ann. Probab. 15 (1987) 344–353], the branching process is subcritical in the sense that it con-verges to a proper limit law. We complement this result by a strong law of large numbers and a central limit theorem for the partial sums of the process. In addition, we study the asymptotic behavior of oscillations of the branch-ing process, that is, of the random segments between successive times when the extinction occurs and the process starts again with the next wave of the immigration. 1. Introduction and statement of results. In this paper we consider the mul-titype branching process with immigration in a random environment (MBPIRE) introduced by Key in [14]. Broadly speaking, the MBPIRE is a vector-valued ran-dom process Zn = (Z(1)n,Z(2)n,...,Z(d)n) that describes evolution of a population of particles of d different types in discrete time n = 0,1,.... The integer numbe