On asymptotics of exchangeable coalescents with multiple collisions

TL;DR

This paper analyzes the asymptotic behavior of the number of collisions in exchangeable coalescents with multiple collisions, deriving various limiting laws and applying the results to specific beta distribution cases.

Contribution

It introduces a coupling technique to determine the limiting distributions of collision counts in $ ext{Lambda}$-coalescents, including normal, stable, and Mittag-Leffler laws, expanding understanding of their asymptotics.

Findings

Limiting laws of $X_n$ include normal, stable, and Mittag-Leffler distributions.

Results apply to $ u$ being a beta$(a-2,b)$ distribution with $a>2$, $b>0$.

Asymptotics derived for absorption time, external branch length, and collision count before coalescence.

Abstract

We study the number of collisions $X_n$ of an exchangeable coalescent with multiple collisions ($\Lambda$-coalescent) which starts with $n$ particles and is driven by rates determined by a finite characteristic measure $\nu({\rm d}x)=x^{-2}\Lambda({\rm d}x)$. Via a coupling technique we derive limiting laws of $X_n$, using previous results on regenerative compositions derived from stick-breaking partitions of the unit interval. The possible limiting laws of $X_n$ include normal, stable with index $1\le\alpha<2$ and Mittag-Leffler distributions. The results apply, in particular, to the case when $\nu$ is a beta$(a-2,b)$ distribution with parameters $a>2$ and $b>0$. The approach taken allows to derive asymptotics of three other functionals of the coalescent, the absorption time, the length of an external branch chosen at random from the $n$ external branches, and the number of collision events that occur before the randomly selected external branch coalesces with one of its neighbours.