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

TL;DR

This paper analyzes the moments and distributional limits of the number of collisions in beta(2,b)-coalescents, providing new asymptotic results and convergence properties for this specific case.

Contribution

It offers new asymptotic expansions and convergence results for the number of collisions in beta(2,b)-coalescents, especially addressing the challenging border case a=2.

Findings

$X_n/\mathbb{E}X_n$ converges almost surely to 1

Properly normalized $X_n$ converges in distribution to a standard normal

Provides asymptotic moments for the number of collisions

Abstract

Expansions are provided for the moments of the number of collisions $X_n$ in the $\beta(2,b)$-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 $\beta(a,b)$-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 $a\neq2$.