Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T17:59:39.582Z Has data issue: false hasContentIssue false
  • English
  • Français

Critical Galton–Watson processes by decreasing 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

This paper deals with the Foster–Pakes model for Galton–Watson branching processes allowing immigration whenever the number of particles is 0. In the critical case we investigate the asymptotic behaviour of the probability of non-extinction, of the expectation and of the variance, and obtain different types of limit theorems depending on the temporally-decreasing sizes of the immigrants.

Keywords

ASYMPTOTIC BEHAVIOURPROBABILITY FOR NON-EXTINCTIONMOMENTSLIMIT DISTRIBUTIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 21 , Issue 1 , March 1984 , pp. 22 - 39
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] Athreya, K. and Ney, P. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
[2] Feller, W. (1966) An Introduction to Probability Theory and its Applications, Vol. 2. Wiley, New York.Google Scholar
[3] Foster, J. H. (1971) A limit theorem for a branching process with state-dependent immigration. Ann. Math. Statist. 42, 1773–1776.Google Scholar
[4] Foster, J. H. and Williamson, J. A. (1971) Limit theorems for the Galton–Watson process with time-dependent immigration. Z. Wahrscheinlichkeitsth. 20, 227–235.Google Scholar
[5] Mitov, V. and Yanev, N. M. (1983) Critical branching processes with decreasing state-dependent immigration. C.R. Acad. Bulg. Sci. 36, No. 2.Google Scholar
[6] Loève, M. (1963) Probability Theory, 3rd edn. Van Nostrand, Princeton, NJ.Google Scholar
[7] Pakes, A. G. (1971) A branching process with a state-dependent immigration component. Adv. Appl. Prob. 3, 301–314.Google Scholar
[8] Pakes, A. G. (1975) Some results for non-supercritical Galton–Watson processes with immigration. Math. Biosci. 24, 71–92.Google Scholar
[9] Pakes, A. G. (1978) On the age distribution of a Markov chain. J. Appl. Prob. 15, 65–77.CrossRefGoogle Scholar
[10] Sevastyanov, B. A. (1971) Branching Processes (in Russian). Nauka, Moscow.Google Scholar
[11] Yamazato, M. (1975) Some results on continuous-time branching processes with state-dependent immigration. J. Math. Soc. Japan 27, 479–496.Google Scholar