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A note on models using the branching process with immigration stopped at zero

Published online by Cambridge University Press:  14 July 2016

E. Seneta  and
S. Ta Varé
E. Seneta*
Affiliation:
University of Sydney
S. Ta Varé*
Affiliation:
Colorado State University
*
∗ Postal address: Department of Mathematical Statistics, University of Sydney, NSW 2006, Australia.
∗∗ Postal address: Department of Statistics, Colorado State University, Fort Collins CO 80523, U.S.A.
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Abstract

The Galton-Watson process with immigration which is time-homogeneous but not permitted when the process is in state 0 (so that this state is absorbing) is briefly studied in the subcritical and supercritical cases. Results analogous to those for the ordinary Galton-Watson process are found to hold. Partly-new techniques are required, although known end-results on the standard process with and without immigration are used also. In the subcritical case a new parameter is found to be relevant, replacing to some extent the criticality parameter.

Keywords

GALTON-WATSON PROCESSIMMIGRATIONSUBCRITICALSUPERCRITICALRENEWAL EQUATIONEXTINCTION-TIME DISTRIBUTIONPLASMIDS
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 1 , March 1983 , pp. 11 - 18
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

Research carried out while this author was visiting Colorado State University.

References

[1] Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
[2] Feller, W. (1968) An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. Wiley, New York.Google Scholar
[3] Gardner, R. C., Metcalf, P. and Bergquist, P. L. (1977) Replication, segregation, and the hereditary stability of bacterial plasmids. Unpublished report, University of Auckland.Google Scholar
[4] Heathcote, C. R., Seneta, E. and Vere-Jones, D. (1967) A refinement of two theorems in the theory of branching processes. Theory Prob. Appl. 12, 297–301.CrossRefGoogle Scholar
[5] Novick, R. P. and Hoppensteadt, F. C. (1978) On plasmid incompatibility. Plasmid 1, 421–434.CrossRefGoogle ScholarPubMed
[6] Pares, A. G. (1971) A branching process with a state dependent immigration component. Adv. Appl. Prob. 3, 301–314.Google Scholar
[7] Seneta, E. (1969) Functional equations and the Galton-Watson process. Adv. Appl. Prob. 1, 1–42.CrossRefGoogle Scholar
[8] Seneta, E. (1970) A note on the supercritical Galton-Watson process with immigration. Math. Biosci. 6, 305–312.CrossRefGoogle Scholar
[9] Seneta, E. (1981) Non-negative Matrices and Markov Chains, 2nd edn. Springer-Verlag, New York.CrossRefGoogle Scholar
[10] 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.CrossRefGoogle Scholar
[11] Zubkov, A. M. (1972) Life periods of a branching process with immigration. Theory Prob. Appl. 17, 174–183.CrossRefGoogle Scholar
[12] Ivanoff, B. G. and Seneta, E. (1983) The critical branching process with immigration. J. Appl. Prob. 20 (4).Google Scholar