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On a theorem of bingham and doney

Published online by Cambridge University Press:  14 July 2016

A. De Meyer
A. De Meyer*
Affiliation:
Katholieke Universiteit Leuven
*
∗Postal address: Mathematical Institute, K. U. Leuven, Celestijnenlaan 200B, 3030 Leuven-Heverlee, Belgium.
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Abstract

We obtain an extension of a theorem of Bingham and Doney connecting the random variables Z1 and W in the supercritical Galton-Watson process. The regular variation of the distribution of Z1 is equivalent to the regular variation of the tail of the distribution of W for integer values of α > 1.

Keywords

GALTON-WATSON PROCESSSUPERCRITICALFUNCTIONAL EQUATIONREGULAR VARIATIONLAPLACE-STIELTJES TRANSFORMII CLASS
Type
Short Communications
Information
Journal of Applied Probability , Volume 19 , Issue 1 , March 1982 , pp. 217 - 220
Copyright
Copyright © Applied Probability Trust 1982 

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References

[1] Athreya, K. B. and Ney, P. E. (1973) Branching Processes. Springer-Verlag, Berlin.Google Scholar
[2] Bingham, N. H. and Doney, R. A. (1974) Asymptotic properties of supercritical branching processes I: The Galton–Watson process. Adv. Appl. Prob. 6, 711–731.CrossRefGoogle Scholar
[3] Bingham, N. H. and Doney, R. A. (1975) Asymptotic properties of supercritical branching processes II: Crump–Mode and Jirina processes. Adv. Appl. Prob. 7, 66–82.Google Scholar
[4] De Haan, L. (1970) On Regular Variation and its Applications to the Weak Convergence of Sample Extremes. Mathematical Centre Tracts 32, Amsterdam.Google Scholar
[5] De Haan, L. (1976) An Abel–Tauber theorem for Laplace transforms. J. London Math. Soc. (2) 17, 102–106.Google Scholar
[6] De Meyer, A. and Teugels, J. L. (1980) On the asymptotic behaviour of the distributions of the busy period and service time in M/G/1. J. Appl. Prob. 17, 802–813.Google Scholar
[7] Stam, A. J. (1973) Regular variation of the tail of a subordinated probability distribution. Adv. Appl. Prob. 5, 308–327.Google Scholar