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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.
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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 

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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