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The simple branching process with infinite mean. I

Published online by Cambridge University Press:  14 July 2016

E. Seneta
E. Seneta*
Affiliation:
Australian National University, Canberra
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Abstract

The simple branching process {Zn} with mean number of offspring per individual infinite, is considered. Conditions under which there exists a sequence {pn} of positive constants such that pn log (1 +Zn) converges in law to a proper limit distribution are given, as is a supplementary condition necessary and sufficient for pn~ constant cn as n→∞, where 0 < c < 1 is a number characteristic of the process. Some properties of the limiting distribution function are discussed; while others (with additional results) are deferred to a sequel.

Keywords

BRANCHING PROCESSGALTON-WATSON PROCESSINFINITE MEANCONVEXITYFUNCTIONAL ITERATION
Type
Short Communications
Information
Journal of Applied Probability , Volume 10 , Issue 1 , March 1973 , pp. 206 - 212
Copyright
Copyright © Applied Probability Trust 1973 

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References

[1] Cramér, H. (1946) Mathematical Methods of Statistics. Princeton University Press, Princeton.Google Scholar
[2] Darling, D. A. (1970) The Galton-Watson process with infinite mean. J. Appl. Prob. 7, 455–456.Google Scholar
[3] Kuczma, M. (1964) Note on Schroder's functional equation. J. Aust. Math. Soc. 4, 149–151.Google Scholar
[4] Kuczma, M. (1966) Sur l'équation fonctionelle de Böttcher. Mathematica (Cluj) 8, 279–285.Google Scholar
[5] Kuczma, M. (1967) Un théorème d'unicité pour l'équation fonctionelle de Böttcher. Mathematica (Cluj) 9, 285–293.Google Scholar
[6] Seneta, E. (1968) On recent theorems concerning the supercritical Galton-Watson process. Ann. Math. Statist. 39, 2098–2102.Google Scholar
[7] Seneta, E. (1969) Functional equations and the Galton-Watson process. Adv. Appl. Prob. 1, 1–42.Google Scholar
[8] Seneta, E. (1969) On Koenigs' ratios for iterates of real functions. J. Aust. Math. Soc. 10, 207–213.CrossRefGoogle Scholar
[9] Szekeres, G. (1958) Regular iteration of real and complex functions. Acta Math. 100, 203–258.Google Scholar