On the total length of external branches for beta-coalescents

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 length of an initial sample of n individuals. For this class of coalescents, it has been proved that n α-1 T (n) →D T, where T (n) is the length of an external branch chosen at random and T is a known nonnegative random variable. For beta(2 − α, α)-coalescents with 1 < α < 2, we obtain lim n→+∞ n 3α-

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

We consider a ${\Lambda}$-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. ${\Lambda}={\delta}_0$, 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 ${\Lambda}$, including Beta$(2-{\alpha},{\alpha})$ with $0<\alpha<1$, M{\"o}hle has proved asymptotics of $L^{(n)}_{ext}$. Here we consider the case when the measure ${\Lambda}$ is Beta$(2-{\alpha},{\alpha})$, with $1<\alpha<2$. We prove that $n^{{\alpha}-2}L^{(n)}_{ext}$ converges in $L^2$ to $\alpha(\alpha-1)\Gamma(\alpha)$. As a consequence, we get that $L^{(n)}_{ext}/L^{(n)}$ converges in probability to $2-\alpha$. 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

On asymptotics of the beta-coalescents

We show that the total number of collisions in the exchangeable coalescent process driven by the beta $(1,b)$ measure converges in distribution to a 1-stable law, as the initial number of particles goes to infinity. The stable limit law is also shown for the total branch length of the coalescent tree. These results were known previously for the instance $b=1$, which corresponds to the Bolthausen--Sznitman coalescent. The approach we take is based on estimating the quality of a renewal approximation to the coalescent in terms of a suitable Wasserstein distance. Application of the method to beta $(a,b)$-coalescents with $0<a<1$ leads to a simplified derivation of the known $(2-a)$-stable limit. We furthermore derive asymptotic expansions for the moments of the number of collisions and of the total branch length for the beta $(1,b)$-coalescent by exploiting the method of sequential approximations.Comment: 25 pages, submitted for publicatio