Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T06:54:10.082Z Has data issue: false hasContentIssue false
  • English
  • Français

Bellman–Harris branching processes with state-dependent immigration

Published online by Cambridge University Press:  14 July 2016

K. V. Mitov  and
N. M. Yanev
K. V. Mitov*
Affiliation:
Institute of Mathematics, Sofia
N. M. Yanev*
Affiliation:
Institute of Mathematics, Sofia
*
∗ Postal address: Department of Probability and Statistics, Institute of Mathematics, Bulgarian Academy of Sciences, 1090 Sofia, P.O. Box 373, Bulgaria.
∗ Postal address: Department of Probability and Statistics, Institute of Mathematics, Bulgarian Academy of Sciences, 1090 Sofia, P.O. Box 373, Bulgaria.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider critical Bellman-Harris processes which admit an immigration component only in the state 0. The asymptotic behaviour of the probability of extinction and of the first two moments is investigated and a limit theorem is also proved.

Keywords

ASYMPTOTIC BEHAVIOURPROBABILITY OF EXTINCTIONMOMENTSLIMIT DISTRIBUTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 22 , Issue 4 , December 1985 , pp. 757 - 765
Copyright
Copyright © Applied Probability Trust 1985 

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.)

References

Athreya, K. and Ney, P. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
Erickson, K. B. (1970) Strong renewal theorem with infinite mean. Trans. Amer. Math. Soc. 151, 263–291.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications , Vol. 2, Wiley, New York.Google Scholar
Foster, J. H. (1971) A limit theorem for a branching process with state-dependent immigration. Ann. Math. Statist. 42, 1773–1776.Google Scholar
Mitov, K. V. (1983) Multitype branching process with immigration in the state zero. Proc. 12th Spring Conf. Union Bulgar. Math., Sunny Beach, 6-9 April 1983 , 202–207 (in Russian).Google Scholar
Mitov, K. V. and Yanev, N. M. (1983) Critical branching processes with decreasing state-dependent immigration. C. R. Acad. Bulgar. Sci. 36, 193–196.Google Scholar
Mitov, K. V. and Yanev, N. M. (1984) Critical Galton-Watson processes with decreasing state-dependent immigration. J. Appl. Prob. 21, 22–39.Google Scholar
Mitov, K. V., Vatutin, B. A. and Yanev, N. M. (1984) Continuous-time branching processes with decreasing state-dependent immigration. Adv. Appl. Prob. 16, 697–714.Google Scholar
Pakes, A. G. (1971) A branching process with a state-dependent immigration component. Adv. Appl. Prob. 3, 301–314.Google Scholar
Pakes, A. G. (1975) Some results for non-supercritical Galton-Watson processes with immigration. Math. Biosci. 24, 71–92.Google Scholar
Pakes, A. G. (1978) On the age distribution of a Markov chain. J. Appl. Prob. 15, 65–77.Google Scholar
Sevastyanov, B. A. (1971) Branching Processes (in Russian). Nauka, Moscow.Google Scholar
Yamazato, M. (1975) Some results on continuous-time branching processes with statedependent immigration. J. Math. Soc. Japan 27, 479–496.Google Scholar