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The Critical Galton-Watson Process Without Further Power Moments

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  14 July 2016

S. V. Nagaev  and
V. Wachtel
S. V. Nagaev*
Affiliation:
Sobolev Institute for Mathematics
V. Wachtel*
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics
*
∗Postal address: Sobolev Institute for Mathematics, Prospect Akademika Koptjuga 4, 630090 Novosibirsk, Russia.
∗∗Postal address: Technische Universität München, Zentrum Mathematik, Bereich M5, TU München, 85747 Garching, Germany. Email address: [email protected]
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Abstract

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In this paper we prove a conditional limit theorem for a critical Galton-Watson branching process {Zn; n ≥ 0} with offspring generating function s + (1 − s)L((1 − s)−1), where L(x) is slowly varying. In contrast to a well-known theorem of Slack (1968), (1972) we use a functional normalization, which gives an exponential limit. We also give an alternative proof of Sze's (1976) result on the asymptotic behavior of the nonextinction probability.

Keywords

Critical Galton-Watson processconditional theoremslowly varying function

MSC classification

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F05: Central limit and other weak theorems
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 3 , September 2007 , pp. 753 - 769
Copyright
Copyright © Applied Probability Trust 2007 

References

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