On the total length of external branches for beta-coalescents

Abstract

In this paper we consider the beta(2 − α, α)-coalescents with 1 < α < 2 and study the moments of external branches, in particular, the total external branch lengthof an initial sample ofnindividuals. For this class of coalescents, it has been proved thatnα-1T(n)→DT, whereT(n)is the length of an external branch chosen at random andTis a known nonnegative random variable. For beta(2 − α, α)-coalescents with 1 < α < 2, we obtain limn→+∞n3α-5𝔼(Lext(n)−n2-α𝔼T)2= ((α − 1)Γ(α + 1))2Γ(4 − α) / ((3 − α)Γ(4 − 2α)).