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On distribution tails and expectations of maxima in critical branching processes

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

K. A. Borovkov  and
V A. Vatutin
K. A. Borovkov*
Affiliation:
Steklov Mathematical Institute
V A. Vatutin*
Affiliation:
Steklov Mathematical Institute
*
∗Postal address for both authors: Steklov Mathematical Institute, Vavilov st. 42, 117966 Moscow GSP-1, Russia.
∗Postal address for both authors: Steklov Mathematical Institute, Vavilov st. 42, 117966 Moscow GSP-1, Russia.
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Abstract

We derive the limit behaviour of the distribution tail of the global maximum of a critical Galton–Watson process and also of the expectations of partial maxima of the process, when the offspring law belongs to the domain of attraction of a stable law. Thus the Lindvall (1976) and Athreya (1988) results are extended to the infinite variance case. It is shown that in the general case these two asymptotics are closely related to each other, and the latter follows readily from the former. We also discuss a related problem from the theory of general branching processes.

Keywords

BRANCHING PROCESSMAXIMA OF GALTON–WATSON PROCESSESINFINITE VARIANCEGENERAL BRANCHING PROCESS

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 3 , September 1996 , pp. 614 - 622
Copyright
Copyright © Applied Probability Trust 1996 

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Footnotes

Research supported by the International Science Foundation Grant MQP000.

V. A. Vatutin was partially supported by the Russian Fundamental Science Foundation Grant no. 93-01–01443.

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