On the length of an external branch in the Beta-coalescent

TL;DR

This paper investigates the asymptotic behavior of the length of external branches in Beta-coalescents, establishing convergence results and providing limit distributions for large sample sizes.

Contribution

It introduces new asymptotic results for external branch lengths in Beta-coalescents, including convergence and limit distributions as the sample size grows.

Findings

Proves convergence of scaled external branch length $n^{\

Provides asymptotics for the number of collisions in the coalescent process.

Derives asymptotics for the block counting process in the $n$-coalescent.

Abstract

In this paper, we consider Beta$(2-{\alpha},{\alpha})$ (with $1<{\alpha}<2$) and related ${\Lambda}$-coalescents. If $T^{(n)}$ denotes the length of an external branch of the $n$-coalescent, we prove the convergence of $n^{{\alpha}-1}T^{(n)}$ when $n$ tends to $ \infty $, and give the limit. To this aim, we give asymptotics for the number $\sigma^{(n)}$ of collisions which occur in the $n$-coalescent until the end of the chosen external branch, and for the block counting process associated with the $n$-coalescent.