Site Frequency Spectrum of the Bolthausen-Sznitman Coalescent

TL;DR

This paper provides explicit formulas and approximations for the site frequency spectrum of the Bolthausen-Sznitman coalescent, along with asymptotic results and distributional analysis of internal branch lengths.

Contribution

It introduces new explicit formulas, efficient approximations, and asymptotic results for the site frequency spectrum of the Bolthausen-Sznitman coalescent, based on its recursive tree construction.

Findings

Explicit formulas for the first two moments of the SFS.

Efficient approximations for small sample sizes.

Distribution and convergence results for internal branch lengths.

Abstract

We derive explicit formulas for the two first moments of he site frequency spectrum $(SFS_{n,b})_{1\leq b\leq n-1}$ of the Bolthausen-Sznitman coalescent along with some precise and efficient approximations, even for small sample sizes $n$. These results provide new $L_2$-asymptotics for some values of $b=o(n)$. We also study the length of internal branches carrying $b>n/2$ individuals. In this case we obtain the distribution function and a convergence in law. Our results rely on the random recursive tree construction of the Bolthausen-Sznitman coalescent.