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A multilevel branching model

Published online by Cambridge University Press:  01 July 2016

D. A. Dawson  and
K. J. Hochberg
D. A. Dawson*
Affiliation:
Carleton University
K. J. Hochberg*
Affiliation:
Bar-Ilan University
*
∗Postal address: Department of Mathematics and Statistics, Carleton University, Ottawa, Canada K1S 5B6.
∗∗Postal address: Department of Mathematics and Computer Science, Bar-Ilan University, 52900 Ramat-Gan, Israel.
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Abstract

We consider a dynamic multilevel population or information system. At each level individuals or information units undergo a Galton–Watson-type branching process in which they can be replicated or removed. In addition, a collection of individuals or information units at a given level constitutes an information unit at the next higher level. Each collection of units also undergoes a Galton–Watson branching process, either dying or replicating. In this paper, we represent this multilevel branching model as a measure-valued stochastic process, study its moment structure, identify the limiting continuous-state approximation and analyse the long-time behavior in both non-critical and critical cases. For example, we obtain an asymptotic expression for the extinction probability for the total population mass process and an analogue of Yaglom's conditioned limit theorem in the critical case.

Keywords

MEASURE-VALUED PROCESSMOMENT MEASURESEXTINCTION PROBABILITYCONDITIONAL LIMIT LAW
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 4 , December 1991 , pp. 701 - 715
Copyright
Copyright © Applied Probability Trust 1991 

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Footnotes

Research partially supported by a Natural Sciences and Engineering Research Council of Canada grant.

Research partially supported by US National Security Agency grant MDA 904-88-H-2044 and US National Science Foundation grant DMS-8800289.

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