Advances in Applied Probability

On asymptotics of the beta coalescents

Alexander Gnedin, Alexander Iksanov, Alexander Marynych, and Martin Möhle

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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.

Article information

Adv. in Appl. Probab., Volume 46, Number 2 (2014), 496-515.

First available in Project Euclid: 29 May 2014

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Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 60G09: Exchangeability
Secondary: 60F05: Central limit and other weak theorems 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

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


Gnedin, Alexander; Iksanov, Alexander; Marynych, Alexander; Möhle, Martin. On asymptotics of the beta coalescents. Adv. in Appl. Probab. 46 (2014), no. 2, 496--515. doi:10.1239/aap/1401369704.

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