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Subcritical Sevastyanov branching processes with nonhomogeneous Poisson immigration

Part of: Markov processes Applications Limit theorems

Published online by Cambridge University Press:  22 June 2017

Ollivier Hyrien ,
Kosto V. Mitov  and
Nikolay M. Yanev
Ollivier Hyrien*
Affiliation:
Fred Hutchinson Cancer Research Center
Kosto V. Mitov*
Affiliation:
Vasil Levski National Military University
Nikolay M. Yanev*
Affiliation:
Bulgarian Academy of Sciences
*
* Postal address: Program in Biostatistics, Bioinformatics, and Epidemiology, Vaccine and Infectious Disease Division, Fred Hutchinson Cancer Research Center, 1100 Fairview Ave. N., Seattle, WA 98109, USA. Email address: ohyrien@fredhutch.org
** Postal address: Faculty of Aviation, Vasil Levski National Military University, 5856 D. Mitropolia, Pleven, Bulgaria. Email address: kmitov@yahoo.com
*** Postal address: Department of Operations Research, Probability and Statistics, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria. Email address: yanev@math.bas.bg
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Abstract

We consider a class of Sevastyanov branching processes with nonhomogeneous Poisson immigration. These processes relax the assumption required by the Bellman–Harris process which imposes the lifespan and offspring of each individual to be independent. They find applications in studies of the dynamics of cell populations. In this paper we focus on the subcritical case and examine asymptotic properties of the process. We establish limit theorems, which generalize classical results due to Sevastyanov and others. Our key findings include a novel law of large numbers and a central limit theorem which emerge from the nonhomogeneity of the immigration process.

Keywords

Branching processimmigrationPoisson processlimit theorem

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F05: Central limit and other weak theorems 60J85: Applications of branching processes 62P10: Applications to biology and medical sciences
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 2 , June 2017 , pp. 569 - 587
Copyright
Copyright © Applied Probability Trust 2017 

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