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The maximum in critical Galton–Watson and birth and death processes

Published online by Cambridge University Press:  14 July 2016

K. Kämmerle  and
H.-J. Schuh
K. Kämmerle
Affiliation:
Johannes Gutenberg-Universität Mainz
H.-J. Schuh*
Affiliation:
Johannes Gutenberg-Universität Mainz
*
∗Postal address: Johannes Gutenberg-Universität Mainz, Fachbereich 17 Mathematik, Saarstr. 21, 6500 Mainz, W. Germany.
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Abstract

In this paper recent results by Weiner [10] on Mn:= max{Z0, · ··, Zn} are strengthened and generalized, where (Zn)n is a critical Galton–Watson branching process with finite and positive offspring variance and Z0 ≡ 1. It is shown that Explicit asymptotic bounds are given for with . If (Zn)n has a linear fractional offspring distribution, it can be embedded in a critical birth and death process (Ẑ t)t. Using martingale methods one obtains thereof.

These results generalize to the case Z0 ≡ k.

Keywords

STOPPING TIMESTRONG MARKOV PROPERTYMARTINGALE INEQUALITIESOPTIONAL STOPPING THEOREM
Type
Research Paper
Information
Journal of Applied Probability , Volume 23 , Issue 3 , September 1986 , pp. 601 - 613
Copyright
Copyright © Applied Probability Trust 1986 

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References

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