On the external branches of coalescents with multiple collisions

TL;DR

This paper develops a recursion for joint moments of external branch lengths in $ ext{Lambda}$-coalescents, derives asymptotic behaviors for the Bolthausen-Sznitman coalescent, and reveals correlation patterns between external branches.

Contribution

It introduces a recursion for joint moments, derives asymptotic results, and provides exact solutions for the Bolthausen-Sznitman coalescent's external branch lengths.

Findings

External branch lengths are positively correlated in the Bolthausen-Sznitman coalescent.

External branch lengths are negatively correlated in the Kingman coalescent for $n \\ge 4$.

Asymptotic joint moments are obtained via a differential equation approach.

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 $n\ge 4$.