Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T14:45:48.975Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential growth of a branching process usually implies stable age distribution

Published online by Cambridge University Press:  14 July 2016

Bo Berndtsson  and
Peter Jagers
Bo Berndtsson*
Affiliation:
Chalmers University of Technology and University of Göteborg
Peter Jagers*
Affiliation:
Chalmers University of Technology and University of Göteborg
*
∗ Postal address: Department of Mathematics, Chalmers University of Technology and University of Göteborg, Fack, S–412 96 Göteborg, Sweden.
∗ Postal address: Department of Mathematics, Chalmers University of Technology and University of Göteborg, Fack, S–412 96 Göteborg, Sweden.
Get access Rights & Permissions [Opens in a new window]

Abstract

Start a Bellman–Harris branching process from one or several ancestors, whose ages are identically distributed random variables. Assume that the life-length distribution decays more quickly than exponentially and that the distribution of ages at start does not give too much mass to high ages (in a sense to be made precise). Then, if the expected population size is an exponential function of time, the ages must follow the stable age distribution of the process.

Keywords

BRANCHING PROCESSSTABLE AGE DISTRIBUTIONCONVOLUTIONBOUNDED ANALYTIC FUNCTION
Type
Short Communications
Information
Journal of Applied Probability , Volume 16 , Issue 3 , September 1979 , pp. 651 - 656
Copyright
Copyright © Applied Probability Trust 1979 

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.)

References

[1] Boas, R. P. (1954) Entire Functions. Academic Press, New York.Google Scholar
[2] Hayman, W. K. (1956) Questions of regularity connected with the Phragmén–Lindelöf principle. I. Math. Pures Appl. 35, 115–126.Google Scholar
[3] Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, London.Google Scholar
[4] Jagers, P. (1978) Balanced exponential growth: What does it mean and when is it there? In Biomathematics and Cell Kinetics, ed. Valleron, A.J. Elsevier North-Holland, Amsterdam.Google Scholar
[5] Pólya, G. (1949) Remarks on characteristic functions. Proc. Berkeley Symp. Math. Statist. Prob., ed. Pólya, G. 115–123.Google Scholar