Translator Disclaimer
June 2014 On asymptotics of the beta coalescents
Alexander Gnedin, Alexander Iksanov, Alexander Marynych, Martin Möhle
Author Affiliations +
Adv. in Appl. Probab. 46(2): 496-515 (June 2014). DOI: 10.1239/aap/1401369704

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 ∞. 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.

Citation

Download Citation

Alexander Gnedin. Alexander Iksanov. Alexander Marynych. Martin Möhle. "On asymptotics of the beta coalescents." Adv. in Appl. Probab. 46 (2) 496 - 515, June 2014. https://doi.org/10.1239/aap/1401369704

Information

Published: June 2014
First available in Project Euclid: 29 May 2014

zbMATH: 1323.60021
MathSciNet: MR3215543
Digital Object Identifier: 10.1239/aap/1401369704

Subjects:
Primary: 60C05 , 60G09
Secondary: 60F05 , 60J10

Keywords: Absorption time , asymptotic expansion , beta coalescent , coupling , number of collisions , total branch length , Wasserstein distance

Rights: Copyright © 2014 Applied Probability Trust

JOURNAL ARTICLE
20 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.46 • No. 2 • June 2014
Back to Top