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A representation for the limiting random variable of a branching process with infinite mean and some related problems

Published online by Cambridge University Press:  14 July 2016

Harry Cohn  and
Anthony G. Pakes
Harry Cohn*
Affiliation:
The Australian National University
Anthony G. Pakes*
Affiliation:
Monash University, Clayton, Victoria
*
∗Now at the University of Melbourne, Parkville, Victoria.
∗∗Research carried out while the author was on leave at Princeton University.
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Abstract

It is known that for a Bienaymé– Galton–Watson process {Zn} whose mean m satisfies 1 < m < ∞, the limiting random variable in the strong limit theorem can be represented as a random sum of i.i.d. random variables and hence that convergence rate results follow from a random sum central limit theorem.

This paper develops an analogous theory for the case m = ∞ which replaces ‘sum' by ‘maximum'. In particular we obtain convergence rate results involving a limiting extreme value distribution. An associated estimation problem is considered.

Keywords

BRANCHING PROCESSESINFINITE MEANSLOW VARIATIONPOINCARÉ FUNCTIONAL EQUATIONCONVERGENCE RATEEXTREME VALUE
Type
Research Papers
Information
Journal of Applied Probability , Volume 15 , Issue 2 , June 1978 , pp. 225 - 234
Copyright
Copyright © Applied Probability Trust 1978 

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