The joint fluctuations of the lengths of the Beta$(2-\alpha, \alpha)$-coalescents

TL;DR

This paper studies the joint fluctuations of branch lengths in Beta$(2- ext{alpha}, ext{alpha})$-coalescent trees with many leaves, showing convergence to a multivariate stable distribution for fixed order lengths.

Contribution

It establishes the asymptotic distribution of the lengths of branches of fixed order in Beta-coalescents, revealing their joint stable fluctuation behavior as the number of leaves grows.

Findings

Joint fluctuations of branch lengths converge to a multivariate stable distribution.

Asymptotic behavior holds for fixed branch orders as leaves tend to infinity.

Results extend understanding of coalescent tree structures in population genetics.

Abstract

We consider Beta$(2-\alpha, \alpha)$-coalescents with parameter range $1 <\alpha<2$ starting from $n$ leaves. The length $\ell^{(n)}_r$ of order $r$ in the $n$-Beta$(2-\alpha, \alpha)$-coalescent tree is defined as the sum of the lengths of all branches that carry a subtree with $r$ leaves. We show that for any $s \in \mathbb N$ the vector of suitably centered and rescaled lengths of orders $1\le r \le s$ converges in distribution to a multivariate stable distribution as the number of leaves tends to infinity.