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The Total External Branch Length of Beta-Coalescents†

Published online by Cambridge University Press:  10 July 2014

IULIA DAHMER ,
GÖTZ KERSTING  and
ANTON WAKOLBINGER
IULIA DAHMER
Affiliation:
Institut für Mathematik, Goethe-Universität, 60054 Frankfurt am Main, Germany (e-mail: dahmer@math.uni-frankfurt.de, kersting@math.uni-frankfurt.de, wakolbinger@math.uni-frankfurt.de)
GÖTZ KERSTING
Affiliation:
Institut für Mathematik, Goethe-Universität, 60054 Frankfurt am Main, Germany (e-mail: dahmer@math.uni-frankfurt.de, kersting@math.uni-frankfurt.de, wakolbinger@math.uni-frankfurt.de)
ANTON WAKOLBINGER
Affiliation:
Institut für Mathematik, Goethe-Universität, 60054 Frankfurt am Main, Germany (e-mail: dahmer@math.uni-frankfurt.de, kersting@math.uni-frankfurt.de, wakolbinger@math.uni-frankfurt.de)
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Abstract

For 1 < α < 2 we derive the asymptotic distribution of the total length of external branches of a Beta(2 − α, α)-coalescent as the number n of leaves becomes large. It turns out that the fluctuations of the external branch length follow those of τn2−α over the entire parameter regime, where τn denotes the random number of coalescences that bring the n lineages down to one. This is in contrast to the fluctuation behaviour of the total branch length, which exhibits a transition at $\alpha_0 = (1+\sqrt 5)/2$ ([18]).

Keywords

Primary 60K35Secondary 60F0560J10
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 6: Honouring the Memory of Philippe Flajolet - Part 2 , November 2014 , pp. 1010 - 1027
Copyright
Copyright © Cambridge University Press 2014 

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Footnotes

†

Work partially supported by the DFG Priority Programme SPP 1590 ‘Probabilistic Structures in Evolution’.

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