On the external branches of coalescents with multiple collisions

2012/09/15 by Dhersin, Jean-Stephane, Moehle, Martin
#FOS: Biological sciences #FOS: Mathematics #Populations and Evolution (q-bio.PE) #Probability (math.PR)

paper · doi:10.48550/arxiv.1209.3380

Abstract

A recursion for the joint moments of the external branch lengths for coalescents with multiple collisions (Λ-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.

Related

• The collision spectrum of Λ-coalescents
• External branch lengths of Λ-coalescents without a dust component
• Tree lengths for general Λ-coalescents and the asymptotic site frequency spectrum around the Bolthausen-Sznitman coalescent
• Asympotic behavior of the total length of external branches for Beta-coalescents
• On asymptotics of exchangeable coalescents with multiple collisions