On the number of collisions in beta(2, $b$)-coalescents

Abstract

Expansions are provided for the moments of the number of collisions $X_n$ in the $β(2,b)$-coalescent restricted to the set $\{1,...,n\}$. We verify that $X_n/\mathbb{E}X_n$ converges almost surely to one and that $X_n$, properly normalized, weakly converges to the standard normal law. These results complement previously known facts concerning the number of collisions in $β(a,b)$-coalescents with $a\in(0,2)$ and $b=1$, and $a>2$ and $b>0$. The case $a=2$ is a kind of `border situation' which seems not to be amenable to approaches used for $a\neq2$.