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A limit theorem for sample maxima and heavy branches in Galton–Watson trees

Published online by Cambridge University Press:  14 July 2016

G. L. O'brien
G. L. O'brien*
Affiliation:
York University
*
∗Postal address: Department of Mathematics, York University, 4700 Keele St., Downsview, Ontario M3J 1P3, Canada.
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Abstract

Let Yn be the maximum of n independent positive random variables with common distribution function F and let Sn be their sum. Then converges to zero in probability if and only if is slowly varying. This result implies that in a supercritical Galton-Watson process which does not become extinct, there cannot be a sequence {τ n} of particles, each descended from the preceding one, such that the fraction of all particles which are descendants of τ n does not converge to zero as n →∞. Weakly m-adic trees, which behave to some extent like sample Galton-Watson trees, can have such sequences of particles.

Keywords

SAMPLE MAXIMUMSAMPLE SUMCONVERGENCE IN PROBABILITYSLOW VARIATIONSUPERCRITICAL GALTON-WATSON TREEWEAKLY m -ADIC
Type
Short Communications
Information
Journal of Applied Probability , Volume 17 , Issue 2 , June 1980 , pp. 539 - 545
Copyright
Copyright © Applied Probability Trust 1980 

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Footnotes

Research supported in part by the Natural Sciences and Engineering Research Council of Canada. I am grateful to Cornell University, whose hospitality I enjoyed while working on part of this paper.

References

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