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On multitype branching processes in a random environment

Published online by Cambridge University Press:  01 July 2016

David Tanny
David Tanny*
Affiliation:
University of Rochester
*
∗Postal address: Department of Mathematics, Mathematical Sciences Building, Rochester, NY 14627, U.S.A.
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Abstract

This paper is concerned with the growth of multitype branching processes in a random environment (mbpre). It is shown that, under suitable regularity conditions, the process either explodes of becomes extinct. A classification theorem is given delineating the cases of explosion or extinction. Furthermore, it is shown that the process grows at an exponential rate on its set of non-extinction provided the process is stable. Criteria is given for non-certain extinction of the mbpre to occur, and an example shows that the stability condition cannot be removed. The method of proof used, in general, is direct probabilistic computation rather than the classical functional iteration techniques. Growth theorems are first proved for increasing mbpre and subsequently transferred to general mbpre using the associated mbpre and the reduced mbpre.

Keywords

MULTITYPE BRANCHING PROCESS IN RANDOM ENVIRONMENTREDUCED BRANCHING PROCESSASSOCIATED BRANCHING PROCESSSTABLE PROCESSREGULAR PROCESSCONDITION Q
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 3 , September 1981 , pp. 464 - 497
Copyright
Copyright © Applied Probability Trust 1981 

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Footnotes

Partially supported by NSF Grant MCS78–02142.

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