On the external branches of coalescents with multiple collisions

Jean-Stéphane Dhersin (Sorbonne Paris Cité)
Martin Möhle (Eberhard Karls Universität Tübingen)

Abstract


A recursion for the joint moments of the external branch lengths for coalescents with multiple collisions (Lambda-coalescents) is provided. This recursion is used to derive asymptotic results as the sample size n tends to infinity for the joint moments of the external branch lengths and for the moments of the total external branch length of the Bolthausen-Sznitman coalescent. These asymptotic results are based on a differential equation approach, which is as well useful to obtain exact solutions for the joint moments of the external branch lengths for the Bolthausen-Sznitman coalescent. The results for example show that the lengths of two randomly chosen external branches are positively correlated for the Bolthausen-Sznitman coalescent, whereas they are negatively correlated for the Kingman coalescent provided that n >= 4.

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Pages: 1-11

Publication Date: March 20, 2013

DOI: 10.1214/EJP.v18-2286

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