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On Asymptotics of the Beta Coalescents

Part of: Markov processes Stochastic processes Combinatorial probability Limit theorems

Published online by Cambridge University Press:  22 February 2016

Alexander Gnedin ,
Alexander Iksanov ,
Alexander Marynych  and
Martin Möhle
Alexander Gnedin*
Affiliation:
University of London
Alexander Iksanov*
Affiliation:
National Taras Shevchenko University of Kyiv
Alexander Marynych*
Affiliation:
National Taras Shevchenko University of Kyiv
Martin Möhle*
Affiliation:
University of Tübingen
*
∗ Postal address: School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK. Email address: a.gnedin@qmul.ac.uk
∗∗ Postal address: Faculty of Cybernetics, National Taras Shevchenko University of Kyiv, 01601 Kyiv, Ukraine. Email address: iksan@univ.kiev.ua
∗∗∗ Postal address: National Taras Shevchenko University of Kyiv, 01601 Kyiv, Ukraine. Email address: marynych@unicyb.kiev.ua
∗∗∗∗ Postal address: Mathematical Institute, University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany. Email address: martin.moehle@uni-tuebingen.de
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Abstract

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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.

Keywords

Absorption timeasymptotic expansionbeta coalescentcouplingnumber of collisionstotal branch lengthWasserstein distance

MSC classification

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)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 2 , June 2014 , pp. 496 - 515
Copyright
© Applied Probability Trust 

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