The asymptotic distribution of the length of Beta-coalescent trees

TL;DR

This paper analyzes the asymptotic behavior of the total branch length in Beta-coalescent trees for different alpha regimes, revealing stable and shifted limit distributions, and also derives the distribution of segregation sites.

Contribution

It provides the first detailed asymptotic distribution results for the total length of Beta-coalescent trees across different alpha regimes.

Findings

For alpha ≤ 1/2(1+√5), the rescaled total length converges to a stable distribution of index alpha.

For alpha > 1/2(1+√5), the total length, after shifting, converges to a nondegenerate limit distribution.

The limit distribution of the number of segregation sites is also derived.

Abstract

We derive the asymptotic distribution of the total length $L_n$ of a $\operatorname {Beta}(2-\alpha,\alpha)$-coalescent tree for $1<\alpha<2$, starting from $n$ individuals. There are two regimes: If $\alpha\le1/2(1+\sqrt{5})$, then $L_n$ suitably rescaled has a stable limit distribution of index $\alpha$. Otherwise $L_n$ just has to be shifted by a constant (depending on $n$) to get convergence to a nondegenerate limit distribution. As a consequence, we obtain the limit distribution of the number $S_n$ of segregation sites. These are points (mutations), which are placed on the tree's branches according to a Poisson point process with constant rate.