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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

Information

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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References

Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer Verlag, Berlin.Google Scholar
Brown, B. M. (1976) Private communication.Google Scholar
Bühler, W. (1969) Ein zentraler Grenzwertsatz für Verzweigungsprozesse. Z. Wahrscheinlichkeitsth. 11, 139–141.CrossRefGoogle Scholar
Cohn, H. (1977) Almost sure convergence of branching processes. Z. Wahrscheinlichkeitsth. 38, 73–81.Google Scholar
Darling, D. A. (1970) The Galton–Watson process with infinite mean. J. Appl. Prob. 7, 455–456.Google Scholar
De Haan, L. (1970) On regular variation and its application to the weak convergence of sample extremes. Math. Centrum, Amsterdam.Google Scholar
Heyde, C. C. (1970a) Extension of a result by Seneta for the supercritical branching process. Ann. Math. Statist. 41, 739–742.Google Scholar
Heyde, C. C. (1970b) A rate of convergence result for the supercritical Galton–Watson process. J. Appl. Prob. 7, 451–454.Google Scholar
Gnedenko, B. V. (1943) Sur la distribution limite du terme maximum d'une série aléatoire. Ann. Math. 44, 169–176.CrossRefGoogle Scholar
Grey, D. (1977) Almost sure convergence in Markov branching processes with infinite mean. J. Appl. Prob. 14, 702–716.Google Scholar
Hudson, I. L. and Seneta, E. (1977) A note on simple branching processes with infinite mean. J. Appl. Prob. 14, 836–842.Google Scholar
Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, London.Google Scholar
Mejzler, D. G. (1949) On a theorem of B. V. Gnedenko. Trudy Inst. Mat. Akad. Nauk. Ukrain. R.S.R. 12, 31–35 (in Russian).Google Scholar
Schuh, H-J. and Barbour, A. (1977) On the asymptotic behaviour of branching processes with infinite mean. Adv. Appl. Prob. 9, 681–723.CrossRefGoogle Scholar
Seneta, E. (1968) On recent theorems concerning the supercritical Galton–Watson process. Ann. Math. Statist. 39, 2098–2102.Google Scholar
Seneta, E. (1973) The simple branching process with infinite mean. J. Appl. Prob. 10, 206–212.Google Scholar
Seneta, E. (1974a) Characterization by functional equations of branching processes limit laws. In Statistical Distributions in Scientific Work, ed. Patil, G. et al. 3, 249–254. Calgary.Google Scholar
Seneta, E. (1974b) Regularly varying functions in the theory of simple branching processes. Adv. Appl. Prob. 6, 408–420.Google Scholar