Asymptotics of the Minimal Clade Size and Related Functionals of Certain Beta-Coalescents

The Beta(2−α,α) n-coalescent with 1<α<2 is a Markov process taking values in the set of partitions of {1,…,n}. It evolves from the initial value {1},…,{n} by merging (coalescing) blocks together into one and finally reaching the absorbing state {1,…,n}. This article aims to give the asymptotic distribution of the size of the minimal clade of 1, which is the block containing 1 at the time of coalescence of the singleton {1}. To this, we express it as a function of the coalescence time of {1}, the number of blocks involved and their sizes. The asymptotic behaviours of those related functionals are therefore also studied

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 the length of an external branch in the Beta-coalescent

In this paper, we consider Beta$(2-{\alpha},{\alpha})$ (with $1<{\alpha}<2$) and related ${\Lambda}$-coalescents. If $T^{(n)}$ denotes the length of an external branch of the $n$-coalescent, we prove the convergence of $n^{{\alpha}-1}T^{(n)}$ when $n$ tends to $\infty$, and give the limit. To this aim, we give asymptotics for the number $\sigma^{(n)}$ of collisions which occur in the $n$-coalescent until the end of the chosen external branch, and for the block counting process associated with the $n$-coalescent