Asymptotics of the minimal clade size and related functionals of certain beta-coalescents

Abstract: This article shows the asymptotics of distributions of various functionals of the Beta$(2-\alpha,\alpha)$ $n$-coalescent process with $1<\alpha<2$ when $n$ goes to infinity. This process is a Markov process taking {values} in the set of partitions of ${1, \dots, n}$, evolving from the intial value ${1},\cdots, {n}$ by merging (coalescing) blocks together into one and finally reaching the absorbing state ${1, \dots, n}$. The minimal clade of $1$ is the block which contains $1$ at the time of coalescence of the singleton ${1}$. The limit size of the minimal clade of $1$ is provided. To this, we express it as a function of the coalescence time of ${1}$ and sizes of blocks at that time. Another quantity concerning the size of the largest block (at deterministic small time and at the coalescence time of ${1}$) is also studied.