Second order behavior of the block counting process of beta coalescents

TL;DR

This paper investigates the small-time behavior of the number of blocks in Beta coalescents, providing a simplified proof of a central limit theorem using coupling with branching processes.

Contribution

It offers a straightforward proof of the functional central limit theorem for the block counting process in Beta coalescents, enhancing understanding of their small-time dynamics.

Findings

Established a simple proof for the CLT of block counts

Connected Beta coalescents with continuous-time branching processes

Confirmed the asymptotic normality of the block counting process

Abstract

The Beta coalescents are stochastic processes modeling the genealogy of a population. They appear as the rescaled limits of the genealogical trees of numerous stochastic population models. In this article, we take interest in the number of blocs at small times in the Beta coalescent. Berestycki, Berestycki and Schweinsberg proved a law of large numbers for this quantity. Recently, Limic and Talarczyk proved that a functional central limit theorem holds as well. We give here a simple proof for an unidimensional version of this result, using a coupling between Beta coalescents and continuous-time branching processes.