Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T03:38:54.270Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuous-time branching processes with decreasing state-dependent immigration

Published online by Cambridge University Press:  01 July 2016

K. V. Mitov ,
V. A. Vatutin  and
N. M. Yanev
K. V. Mitov*
Affiliation:
Institute of Mathematics, Sofia
V. A. Vatutin*
Affiliation:
Steklov Mathematical Institute, Moscow
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: Steklov Mathematical Institute, Academy of Sciences of the USSR, 117969 Moscow, 42 Vavilov St, USSR.
∗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

This paper deals with continuous-time branching processes which allow a temporally-decreasing immigration whenever the population size is 0. In the critical case the asymptotic behaviour of the probability of non-extinction and of the first two moments is investigated and different types of limit theorems are also proved.

Keywords

ASYMPTOTIC BEHAVIOURPROBABILITY OF NON-EXTINCTIONMOMENTSLIMIT DISTRIBUTIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 16 , Issue 4 , December 1984 , pp. 697 - 714
Copyright
Copyright © Applied Probability Trust 1984 

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

[1] Asmussen, S. and Hering, H. (1983) Branching Processes. Birkhauser, Boston.CrossRefGoogle Scholar
[2] Athreya, K. and Ney, P. (1972) Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
[3] Badalbaev, I. S. and Rahimov, I. (1978) Critical branching processes with immigration of decreasing intensity (in Russian). Theory Prob. Appl. 23, 275–283.Google Scholar
[4] Feller, W. (1966) An Introduction to Probability Theory and its Applications, Vol. 2. Wiley, New York.Google Scholar
[5] Ford, L. R. (1955) Differential Equations. McGraw-Hill, New York.Google Scholar
[6] Foster, J. H. (1971) A limit theorem for a branching processes with state-dependent immigration. Ann. Math. Statist. 42, 1773–1776.CrossRefGoogle Scholar
[7] Foster, J. H. and Williamson, J. A. (1971) Limit theorems for the Galton-Watson process with time-dependent immigration. Z. Wahrscheinlichkeitsth. 20, 227–235.CrossRefGoogle Scholar
[8] Hering, H. (1973) Asymptotic behaviour of immigration-branching processes with general set of types. I: Critical branching part. Adv. Appl. Prob. 5, 391–416.Google Scholar
[9] 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
[10] 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
[11] Mitov, K. V. and Yanev, N. M. (1984) Branching processes with decreasing state-dependent immigration. Serdica (Sofia) 10.Google Scholar
[12] Pakes, A. G. (1971) A branching process with a state-dependent immigration component. Adv. Appl. Prob. 3, 301–314.Google Scholar
[13] Pakes, A. G. (1974) On supercritical Galton-Watson processes allowing immigration. J. Appl. Prob. 11, 814–817.CrossRefGoogle Scholar
[14] Pakes, A. G. (1975) On Markov branching processes with immigration. Sankhya A 37, 129–138.Google Scholar
[15] Pakes, A. G. (1975) Some results for non-supercritical Galton-Watson processes with immigration. Math. Biosci. 24, 71–92.CrossRefGoogle Scholar
[16] Pakes, A. G. (1978) On the age distribution of a Markov chain. J. Appl. Prob. 15, 65–77.Google Scholar
[17] Seneta, E. (1976) Regularly Varying Functions. Lecture Notes in Mathematics 508, Springer-Verlag, Berlin.Google Scholar
[18] Sevastyanov, B. A. (1957) Limit theorems for branching processes of special form (in Russian). Theory Prob. Appl. 2, 339–348.Google Scholar
[19] Sevastyanov, B. A. (1971) Branching Processes (in Russian). Nauka, Moscow.Google Scholar
[20] Yamazato, M. (1975) Some results on continuous time branching processes with state-dependent immigration. J. Math. Soc. Japan 27, 479–496.Google Scholar
[21] Yosida, K. (1960) Lectures on Differential and Integral Equations. Interscience, New York.Google Scholar