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


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.


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Alexander Gnedin. Alexander Iksanov. Alexander Marynych. Martin Möhle. "On asymptotics of the beta coalescents." Adv. in Appl. Probab. 46 (2) 496 - 515, June 2014.


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

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

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


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Vol.46 • No. 2 • June 2014
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