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On a result of Rubin and Vere-Jones concerning subcritical branching processes

Published online by Cambridge University Press:  14 July 2016

Fred M. Hoppe
Fred M. Hoppe*
Affiliation:
University of Alberta, Edmonton
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Abstract

If a subcritical Galton-Watson process is initiated with an arbitrary mass distribution, then it is known that under certain conditions proper conditional limit distributions exist, depending on a single parameter. It is shown here that there is a one-to-one correspondence between these distributions and those arising from the process with a linear offspring probability generating function.

Keywords

GALTON-WATSON PROCESSYAGLOM LIMITREGULAR VARIATIONLINEARIZATIONFUNCTIONAL EQUATIONSCONVEXITYGENERATING FUNCTIONS
Type
Short Communications
Information
Journal of Applied Probability , Volume 13 , Issue 4 , December 1976 , pp. 804 - 808
Copyright
Copyright © Applied Probability Trust 1976 

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References

[1] Heathcote, C. R., Seneta, E. and Vere-Jones, D. (1967) A refinement of two theorems in the theory of branching processes. Theor. Prob. Appl. 12, 341–346.Google Scholar
[2] Hoppe, F. (1975a) Functional Equations with Applications to Multitype Galton-Watson Branching Processes. Ph.D. dissertation, Princeton University.Google Scholar
[3] Hoppe, F. (1975b) Stationary measures for multitype branching processes. J. Appl. Prob. 12, 219–227.CrossRefGoogle Scholar
[4] Rubin, H. and Vere-Jones, D. (1968) Domains of attraction for the subcritical Galton-Watson process. J. Appl. Prob. 5, 216–219.Google Scholar
[5] Seneta, E. (1971) On invariant measures for simple branching processes. J. Appl. Prob. 8, 43–51.CrossRefGoogle Scholar
[6] Seneta, E. and Vere-Jones, D. (1966) On quasi-stationary distributions in discrete time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3, 403–434.Google Scholar