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Asymptotic behaviour of continuous time, continuous state-space branching processes

Published online by Cambridge University Press:  14 July 2016

D. R. Grey
D. R. Grey*
Affiliation:
University of Sheffield
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Abstract

Results on the behaviour of Markov branching processes as time goes to infinity, hitherto obtained for models which assume a discrete state-space or discrete time or both, are here generalised to a model with both state-space and time continuous. The results are similar but the methods not always so.

Keywords

BRANCHING PROCESSESCONTINUOUS STATE-SPACECONTINUOUS TIMELIMIT LAWLAPLACE TRANSFORMFUNCTIONAL ITERATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 11 , Issue 4 , December 1974 , pp. 669 - 677
Copyright
Copyright © Applied Probability Trust 1974 

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References

[1]Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
[2]Bingham, N. H. and Doney, R. A. (1974) Asymptotic properties of supercritical branching processes, I: Galton-Watson processes. Adv. Appl. Prob. 6, 711–731.Google Scholar
[3]Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
[4]Jirina, M. (1958) Stochastic branching processes with continuous state space. Czech. Math. J. 8, 292–313.Google Scholar
[5]Lamperti, J. (1967) The limit of a sequence of branching processes. Z. Wahrscheinlichkeitsth. 7, 271–288.Google Scholar
[6]Lamperti, J. (1967) Continuous state branching processes. Bull. Amer. Math. Soc. 73, 382–386.Google Scholar
[7]Seneta, E. (1969) Functional equations and the Galton-Watson process. Adv. Appl. Prob. 1, 1–42.Google Scholar
[8]Seneta, E. (1973) The simple branching process with infinite mean, I. J. Appl. Prob. 10, 206–212.Google Scholar
[9]Seneta, E. and Vere-Jones, D. (1969) On a problem of M. Jirina concerning continuous state branching processes. Czech. Math. J. 19, 277–283.Google Scholar
[10]Silverstein, M. L. (1967–8). A new approach to local times. J. Math. Mech. 17, 10231054.Google Scholar
[11]Watanabe, S. (1968) A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ. 8, 141–167.Google Scholar
[12]Watanabe, S. (1969) On two dimensional Markov processes with branching property. Trans. Amer. Math. Soc. 136, 447–466.Google Scholar