Second order behavior of the block counting process of beta coalescents

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.

9 pages. The result of the previous version was wrong due to a major error. Our method only gives a unidimensional central limit theorem

References in corpus (5)

• Small-time behavior of beta coalescents
• Proof(s) of the Lamperti representation of Continuous-State Branching Processes
• The $\Lambda$-coalescent speed of coming down from infinity
• Second-order asymptotics for the block counting process in a class of regularly varying $\Lambda$-coalescents
• Speed of coming down from infinity for birth and death processes