  We consider a multitype branching process with immigration in a random
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 converges
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 branching
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.
