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On maximum family size in branching processes

Part of: Stochastic processes Limit theorems Markov processes

Published online by Cambridge University Press:  14 July 2016

Ibrahim Rahimov  and
George P. Yanev
Ibrahim Rahimov*
Affiliation:
Uzbek Academy of Sciences
George P. Yanev*
Affiliation:
Bulgarian Academy of Sciences
*
∗Postal address: Box 1339, King Fahd University of Petroleum Minerals, Dhahran 31261, Saudi Arabia.
∗∗Postal address: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Box 373, 1090 Sofia, Bulgaria. Email address: [email protected].
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Abstract

The number Yn of offspring of the most prolific individual in the nth generation of a Bienaymé–Galton–Watson process is studied. The asymptotic behaviour of Yn as n → ∞ may be viewed as an extreme value problem for i.i.d. random variables with random sample size. Limit theorems for both Yn and EYn provided that the offspring mean is finite are obtained using some convergence results for branching processes as well as a transfer limit lemma for maxima. Subcritical, critical and supercritical branching processes are considered separately.

Keywords

Bienaymé–Galton–Watson branching processmax-stabilitymax-semi-stabilityrandom sample sizetransfer theorems

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G70: Extreme value theory; extremal processes 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 3 , September 1999 , pp. 632 - 643
Copyright
Copyright © Applied Probability Trust 1999 

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