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The critical branching process with immigration stopped at zero

Published online by Cambridge University Press:  14 July 2016

B. Gail Ivanoff  and
E. Seneta
B. Gail Ivanoff*
Affiliation:
University of Ottawa
E. Seneta*
Affiliation:
University of Sydney
*
∗Postal address: Department of Mathematics, University of Ottawa, Ont., Canada K1N 9B4.
∗∗Postal address: Department of Mathematical Statistics, University of Sydney, NSW 2006, Australia.
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Abstract

Limit theorems for the Galton–Watson process with immigration (BPI), where immigration is not permitted when the process is in state 0 (so that this state is absorbing), have been studied for the subcritical and supercritical cases by Seneta and Tavaré (1983). It is pointed out here that, apart from a change of context, the corresponding theorem in the critical case has been obtained by Vatutin (1977). Extensions which follow from a more general form of initial distribution are sketched, including a new form of limit result (7).

Keywords

EXTINCTION-TIME DISTRIBUTIONREGULARLY VARYING FUNCTIONABELIAN–TAUBERIAN
Type
Short Communications
Information
Journal of Applied Probability , Volume 22 , Issue 1 , March 1985 , pp. 223 - 227
Copyright
Copyright © Applied Probability Trust 1985 

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References

Feller, W. (1966) An Introduction to Probability Theory and its Applications, Vol. II. Wiley, New York.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. I, 3rd edn. Wiley, New York.Google Scholar
Galambos, J. and Seneta, E. (1973) Regularly varying sequences. Proc. Amer. Math. Soc. 41, 110–116.CrossRefGoogle Scholar
Höpfner, R. (1983) Über einige Klassen von zustandabh?ngigen Galton–Watson-Prozessen. Dissertation, Fachbereich Mathematik, Johannes-Gutenberg-Universit?t in Mainz.Google Scholar
Kesten, H., Ney, P. and Spitzer, F. (1966) The Galton–Watson process with mean one and finite variance. Teor. Veroyatnost. i Primenen. 11, 579–611.Google Scholar
Pakes, A. G. (1975) Some results for non-supercritical Galton–Watson processes with immigration. Math. Biosci. 24, 71–92.CrossRefGoogle Scholar
Parameswaran, S. (1961) Partition functions whose logarithms are slowly oscillating. Trans. Amer. Math. Soc. 100, 217–240.CrossRefGoogle Scholar
Seneta, E. (1970) An explicit-limit theorem for the critical Galton–Watson process with immigration. J. R. Statist. Soc. B 32, 149–152.Google Scholar
Seneta, E. (1973) A Tauberian theorem of E. Landau and W. Feller. Ann. Prob. 1, 1057–1058.Google Scholar
Seneta, E. and Tavaré, S. (1983) A note on models using the branching process with immigration stopped at zero. J. Appl. Prob. 20, 11–18.CrossRefGoogle Scholar
Vatutin, V. A. (1977) A conditional limit theorem for a critical branching process with immigration (in Russian.) Mat. Zemetki 21, 727–736.Google Scholar
Zubkov, A. M. (1972) Life-periods of a branching process with immigration. Theory Prob. Appl. 17, 174–183.Google Scholar