On asymptotics of the beta-coalescents

TL;DR

This paper analyzes the asymptotic behavior of the beta-coalescent process, showing convergence to a stable law for the number of collisions and total branch length, and provides new asymptotic expansions for moments.

Contribution

It introduces a renewal approximation approach to derive stable law limits and asymptotic expansions for beta-coalescents, extending previous results to a broader class.

Findings

Total collisions converge to a 1-stable law as initial particles grow.

Total branch length also converges to a 1-stable law.

Derived asymptotic expansions for moments of collisions and branch length.

Abstract

We show that the total number of collisions in the exchangeable coalescent process driven by the beta $(1,b)$ measure converges in distribution to a 1-stable law, as the initial number of particles goes to infinity. The stable limit law is also shown for the total branch length of the coalescent tree. These results were known previously for the instance $b=1$, which corresponds to the Bolthausen--Sznitman coalescent. The approach we take is based on estimating the quality of a renewal approximation to the coalescent in terms of a suitable Wasserstein distance. Application of the method to beta $(a,b)$-coalescents with $0<a<1$ leads to a simplified derivation of the known $(2-a)$-stable limit. We furthermore derive asymptotic expansions for the moments of the number of collisions and of the total branch length for the beta $(1,b)$-coalescent by exploiting the method of sequential approximations.