The Annals of Probability

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

Alexander Roitershtein

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Abstract

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.

Article information

Source
Ann. Probab., Volume 35, Number 4 (2007), 1573-1592.

Dates
First available in Project Euclid: 8 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1181334253

Digital Object Identifier
doi:10.1214/009117906000001015

Mathematical Reviews number (MathSciNet)
MR2330980

Zentralblatt MATH identifier
1117.60079

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

Keywords
Multitype branching processes with immigration in a random environment strong law of large numbers central limit theorem limiting distribution environment viewed from the particle

Citation

Roitershtein, Alexander. A note on multitype branching processes with immigration in a random environment. Ann. Probab. 35 (2007), no. 4, 1573--1592. doi:10.1214/009117906000001015. https://projecteuclid.org/euclid.aop/1181334253


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