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Asymptotic properties of the stationary measure of a Markov branching process

Published online by Cambridge University Press:  14 July 2016

Y. S. Yang
Y. S. Yang*
Affiliation:
Nanyang University, Singapore
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Abstract

The asymptotic properties of the unique stationary measure of a Markov branching process will be given. In the critical case with finite variance, the result can be deduced from a result for discrete time processes of Kesten, Ney and Spitzer (1966) where the proof makes use of a stronger assumption than the finiteness of variance. For the continuous time case where the stationary measure has an explicit form, we can use the discrete renewal theorem which takes care of the infinite variance case as well.

Keywords

MARKOV BRANCHING PROCESSSTATIONARY MEASURESASYMPTOTIC PROPERTIES
Type
Short Communications
Information
Journal of Applied Probability , Volume 10 , Issue 2 , June 1973 , pp. 447 - 450
Copyright
Copyright © Applied Probability Trust 1973 

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References

Harris, T. E. (1963) The Theory of Branching Processes. Springer, Berlin.Google Scholar
Karlin, S. (1968) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
Kesten, H., Ney, P. and Spitzer, F. (1966) The Galton-Watson process with mean one and finite variance. Teor. Veroyat. Primen. 11, 579–611.Google Scholar
Seneta, E. (1971) On invariant measures for simple branching processes. J. Appl. Prob. 8, 43–51.CrossRefGoogle Scholar
Yang, Y. S. (1972) On branching processes allowing immigration. J. Appl. Prob. 9, 24–31.Google Scholar