On the external branches of coalescents with multiple collisions 

Jean-Stephane Dhersin (1) , Martin Moehle (2)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 Mathematisches Institut [Tübingen]

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 $n\ge 4$.

Mots clés: Asymptotic expansions; Bolthausen-Sznitman coalescent; external branches; joint moments; Kingman coalescent; multiple collisions.; multiple collisions

Domaines: Probabilités [math.PR]; Génétique des populations [q-bio.PE]

Jean-Stephane Dhersin, Martin Moehle. On the external branches of coalescents with multiple collisions . 2012. ⟨hal-00732541v3⟩