Hostname: page-component-7f64f4797f-7vssv Total loading time: 0 Render date: 2025-11-04T01:08:03.596Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Information

Type
Short Communications
Information
Journal of Applied Probability , Volume 10 , Issue 1 , March 1973 , pp. 206 - 212
Copyright
Copyright © Applied Probability Trust 1973 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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