Total internal and external lengths of the Bolthausen-Sznitman coalescent

TL;DR

This paper establishes a weak law of large numbers for the total internal length of the Bolthausen-Sznitman coalescent and derives the limiting distribution of the total external length, with applications to population genetics.

Contribution

It introduces new probabilistic results for the lengths in the Bolthausen-Sznitman coalescent, extending previous findings and applying them to genetic mutation analysis.

Findings

Weak law of large numbers for internal length

Weak limit law for external length

Application to total mutations in genealogies

Abstract

In this paper, we study a weak law of large numbers for the total internal length of the Bolthausen-Szmitman coalescent. As a consequence, we obtain the weak limit law of the centered and rescaled total external length. The latter extends results obtained by Dhersin & M\"ohle \cite{DM12}. An application to population genetics dealing with the total number of mutations in the genealogical tree is also given.