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On higher moments of the population size in a subcritical branching process

Published online by Cambridge University Press:  14 July 2016

Eric Willekens
Eric Willekens*
Affiliation:
Catholic University Leuven
*
∗Present address: Department of Mathematics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.
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Abstract

Let {Z(t), t ≧ 0} be an age-dependent subcritical branching process. In this paper we show that if the lifetime distribution is subexponential, EZα (t) ~ EZ(t) (t →∞) for every α ≧ 1. If furthermore the lifetime distribution has a subexponential density, a rate of convergence result in the above relation is established.

Keywords

SUBEXPONENTIAL DISTRIBUTIONS
Type
Short Communications
Information
Journal of Applied Probability , Volume 25 , Issue 2 , June 1988 , pp. 413 - 417
Copyright
Copyright © Applied Probability Trust 1988 

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Footnotes

The author is a research assistant of the Belgian National Fund for Scientific Research.

References

[1] Athreya, K. B. and Ney, P. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
[2] Bellman, R. and Harris, T. E. (1952) On age-dependent binary branching processes. Ann. Math. 55, 280–295.CrossRefGoogle Scholar
[3] Chistyakov, V. (1964) A theorem on sums of independent positive random variables and its applications to branching processes. Theory Prob. Appl. 9, 640–648.Google Scholar
[4] Chover, J., Ney, P. and Wainger, S. (1973) Functions of probability measures. J. d'Anal. Math. 26, 255–302.Google Scholar
[5] Chover, J., Ney, P. and Wainger, S. (1973) Degeneracy properties of subcritical branching processes. Ann. Prob. 1, 663–673.CrossRefGoogle Scholar
[6] Embrechts, P. (1984) Subexponential distribution functions and their applications: a review. Proc. 7th Conf. Probability Theory, Brasov , 125–136.Google Scholar
[7] Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979) Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335–347.Google Scholar
[8] Omey, E. and Willekens, E. (1987) Second order behaviour of the tail of a subordinated probability distribution: finite mean case. Commun. Statist.: Stochastic Models 3, 311–342.Google Scholar
[9] Riordan, J. (1958) An Introduction to Combinatorial Analysis. Wiley, New York.Google Scholar