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On maximum family size in branching processes

Part of: Stochastic processes Markov processes Limit theorems

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: gyanev@tarski.math.usf.edu.
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Abstract

The number Y n of offspring of the most prolific individual in the nth generation of a Bienaymé–Galton–Watson process is studied. The asymptotic behaviour of Y n as n → ∞ may be viewed as an extreme value problem for i.i.d. random variables with random sample size. Limit theorems for both Y n and EY n 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

Information

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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