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A limit theorem for the maxima of the para-critical simple branching process

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  01 July 2016

Anthony G. Pakes
Anthony G. Pakes*
Affiliation:
University of Western Australia
*
∗ Postal address: Department of Mathematics, University of Western Australia, Nedlands, WA, Australia 6907. Email address: pakes@maths.uwa.edu.au
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Abstract

Let Mn denote the size of the largest amongst the first n generations of a simple branching process. It is shown for near critical processes with a finite offspring variance that the law of Mn/n, conditioned on no extinction at or before n, has a non-defective weak limit. The proof uses a conditioned functional limit theorem deriving from the Feller-Lindvall (CB) diffusion limit for branching processes descended from increasing numbers of ancestors. Subsidiary results are given about hitting time laws for CB diffusions and Bessel processes.

Keywords

Branching processmaximum generation sizediffusion limitBessel processspecial functions

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F17: Functional limit theorems; invariance principles 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Secondary: 33C10: Bessel and Airy functions, cylinder functions, $_0F_1$ 33C15: Confluent hypergeometric functions, Whittaker functions, $_1F_1$
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 3 , September 1998 , pp. 740 - 756
Copyright
Copyright © Applied Probability Trust 1998 

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