A note on multitype branching processes with immigration in a random environment. (English) Zbl 1117.60079

Summary: We consider a multitype branching process with immigration in a random environment introduced by E. S. Key in [Ann. Probab. 15, 344–353 (1987; Zbl 0623.60090)]. It was shown by Key that, under the assumptions made in [loc. cit.], 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.


60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K37 Processes in random environments
60F05 Central limit and other weak theorems
60F15 Strong limit theorems


Zbl 0623.60090
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