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On Asymptotics of the Beta Coalescents

Published online by Cambridge University Press:  22 February 2016

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: [email protected]
∗∗ Postal address: Faculty of Cybernetics, National Taras Shevchenko University of Kyiv, 01601 Kyiv, Ukraine. Email address: [email protected]
∗∗∗ Postal address: National Taras Shevchenko University of Kyiv, 01601 Kyiv, Ukraine. Email address: [email protected]
∗∗∗∗ Postal address: Mathematical Institute, University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany. Email address: [email protected]
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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.

Type
General Applied Probability
Copyright
© Applied Probability Trust 

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