Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T06:18:59.205Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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 Unpublished report, University of Auckland.Google Scholar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