Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T07:45:36.775Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

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.)

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