Hostname: page-component-7dd5485656-hw7sx Total loading time: 0 Render date: 2025-10-30T23:19:22.552Z Has data issue: false hasContentIssue false
  • English
  • Français

On branching processes allowing immigration

Published online by Cambridge University Press:  14 July 2016

Y. S. Yang
Y. S. Yang*
Affiliation:
Nanyang University, Singapore
Get access Rights & Permissions [Opens in a new window]

Abstract

Continuous time one-type branching processes allowing immigration are considered. The invariant measure, which is shown to be unique, is exhibited. From this, a condition for positive recurrence similar to that of Heathcote's in the discrete time case is obtained. For the critical discrete time case, Seneta's sufficient condition for positive recurrence is improved to give a necessary and sufficient condition.

Keywords

CONTINUOUS TIME MARKOV CHAINSBRANCHING PROCESSES ALLOWING IMMIGRATIONINVARIANT MEASURESPOSITIVE RECURRENCE

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 9 , Issue 1 , March 1972 , pp. 24 - 31
Copyright
Copyright © Applied Probability Trust 1972 

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.)

Article purchase

Temporarily unavailable

References

[1] Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities. Springer, New York.Google Scholar
[2] Harris, T. E. (1957) Transient Markov chains with stationary measures. Proc. Amer. Math. Soc. 8, 937–942.CrossRefGoogle Scholar
[3] Harris, T. E. (1963) The Theory of Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
[4] Heathcote, C. R. (1965) A branching process allowing immigration. J. R. Statist. Soc. B 11, 1–14.Google Scholar
[5] Heathcote, C. R. (1966) Corrections and comments on the paper “A branching process allowing immigration”. J. R. Statist. Soc. B 28, 213–217.Google Scholar
[6] Heathcote, C. R., Seneta, E. and Vere-Jones, D. (1967) A refinement of two theorems in the theory of branching processes. Teor. Veroyat. Primen. 12, 341–346.Google Scholar
[7] Karlin, S. (1968) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
[8] 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
[9] Seneta, E. (1967) The Galton-Watson process with mean one. J. Appl. Prob. 4, 489–495.Google Scholar
[10] Seneta, E. (1968) The stationary distribution of a branching process allowing immigration: A remark on the critical case. J. R. Statist. Soc. B 30, 176–179.Google Scholar
[11] Seneta, E. (1969) Functional equations and the Galton-Watson process. Adv. Appl. Prob. 1, 1–42.Google Scholar
[12] Seneta, E. (1970) On invariant measures for simple branching process. (Summary). Bull. Aust. Math. Soc. 2, 359–362.Google Scholar