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Asympotic behavior of the total length of external branches for Beta-coalescents

We consider a $Λ$-coalescent and we study the asymptotic behavior of the total length $L^{(n)}_{ext}$ of the external branches of the associated $n$-coalescent. For Kingman coalescent, i.e. $Λ=δ_0$, the result is well known and is useful, together with the total length $L^{(n)}$, for Fu and Li's test of neutrality of mutations% under the infinite sites model asumption . For a large family of measures $Λ$, including Beta$(2-α,α)$ with $0<α<1$, M{ö}hle has proved asymptotics of $L^{(n)}_{ext}$. Here we consider the case when the measure $Λ$ is Beta$(2-α,α)$, with $1<α<2$. We prove that $n^{α-2}L^{(n)}_{ext}$ converges in $L^2$ to $α(α-1)Γ(α)$. As a consequence, we get that $L^{(n)}_{ext}/L^{(n)}$ converges in probability to $2-α$. To prove the asymptotics of $L^{(n)}_{ext}$, we use a recursive construction of the $n$-coalescent by adding individuals one by one. Asymptotics of the distribution of $d$ normalized external branch lengths and a related moment result are also given.