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Extinction of non-homogeneous Galton-Watson processes

Published online by Cambridge University Press:  14 July 2016

Miloslav Jirina
Miloslav Jirina*
Affiliation:
The Flinders University of South Australia
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Abstract

In the paper the problem of the extinction of non-homogeneous Galton-Watson processes with one type of particle is studied. It is proved that the process becomes extinct if the expectations of the process do not converge to a limit (finite or infinite). If the expectations have a finite limit, then simple necessary and sufficient conditions for the extinction are proved. The general case remains open; however two more sufficient conditions which are also necessary under some restrictions are given.

Keywords

NON-HOMOGENEOUS GALTON–WATSON PROCESSESDEGENERACY OR EXTINCTION OF THE PROCESSNECESSARY AND SUFFICIENT CONDITIONS
Type
Short Communications
Information
Journal of Applied Probability , Volume 13 , Issue 1 , March 1976 , pp. 132 - 137
Copyright
Copyright © Applied Probability Trust 1976 

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References

[1] Čistjakov, V. P. and Markova, N. P. (1962) On some theorems for inhomogeneous branching processes. Dokl. Akad. Nauk. SSSR 147, 317–320.Google Scholar
[2] Fearn, D. H. (1972) Galton-Watson processes with generation dependence. Proc. 6th Berkeley Symp. Math. Statist. Prob. 4, 159–172.Google Scholar
[3] Jagers, P. (1974) Galton-Watson processes in varying environments. J. Appl. Prob. 11, 174–178.Google Scholar