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On the total length of external branches for beta-coalescents

Part of: Markov processes

Published online by Cambridge University Press:  21 March 2016

Jean-Stéphane Dhersin  and
Linglong Yuan
Jean-Stéphane Dhersin*
Affiliation:
Université Paris 13
Linglong Yuan*
Affiliation:
Université Paris 13
*
∗ Postal address: Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS, UMR 7539, F-93430, Villetaneuse, France. Email address: dhersin@math.univ-paris13.fr
∗∗ Current address: Department of Mathematics, Uppsala University, PO Box 480, SE-751 06, Uppsala, Sweden. Email address: yuanlinglongcn@gmail.com
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Abstract

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In this paper we consider the beta(2 − α, α)-coalescents with 1 < α < 2 and study the moments of external branches, in particular, the total external branch length of an initial sample of n individuals. For this class of coalescents, it has been proved that nα-1T(n) →DT, where T(n) is the length of an external branch chosen at random and T is a known nonnegative random variable. For beta(2 − α, α)-coalescents with 1 < α < 2, we obtain limn→+∞n3α-5 𝔼(Lext(n) − n2-α𝔼T)2 = ((α − 1)Γ(α + 1))2Γ(4 − α) / ((3 − α)Γ(4 − 2α)).

Keywords

Coalescent processbeta-coalescenttotal external branch lengthFu and Li's statistical test

MSC classification

Primary: 60J28: Applications of continuous-time Markov processes on discrete state spaces 92D25: Population dynamics (general)
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J85: Applications of branching processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 3 , September 2015 , pp. 693 - 714
Copyright
Copyright © Applied Probability Trust 2015 

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