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Multi-type branching processes with time-dependent branching rates

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  16 November 2018

D. Dolgopyat ,
P. Hebbar ,
L. Koralov  and
M. Perlman
D. Dolgopyat*
Affiliation:
Univeristy of Maryland
P. Hebbar*
Affiliation:
Univeristy of Maryland
L. Koralov*
Affiliation:
Univeristy of Maryland
M. Perlman*
Affiliation:
Stanford Univeristy
*
* Postal address: Department of Mathematics, University of Maryland, College Park, MD 20742, USA.
* Postal address: Department of Mathematics, University of Maryland, College Park, MD 20742, USA.
* Postal address: Department of Mathematics, University of Maryland, College Park, MD 20742, USA.
*** Postal address: Department of Mathematics, Stanford University, Stanford, CA 94305, USA.
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Abstract

Under mild nondegeneracy assumptions on branching rates in each generation, we provide a criterion for almost sure extinction of a multi-type branching process with time-dependent branching rates. We also provide a criterion for the total number of particles (conditioned on survival and divided by the expectation of the resulting random variable) to approach an exponential random variable as time goes to ∞.

Keywords

Multi-type branchingextinction probabilitynonnegative matrix productexponential limit law

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F05: Central limit and other weak theorems 60F10: Large deviations
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 3 , September 2018 , pp. 701 - 727
Copyright
Copyright © Applied Probability Trust 2018 

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