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On Λ-Coalescents with Dust Component

Part of: Limit theorems Special processes Markov processes Stochastic processes Combinatorial probability

Published online by Cambridge University Press:  14 July 2016

Alexander Gnedin ,
Alexander Iksanov  and
Alexander Marynych
Alexander Gnedin*
Affiliation:
Utrecht University
Alexander Iksanov*
Affiliation:
National Taras Shevchenko University of Kiev
Alexander Marynych*
Affiliation:
National Taras Shevchenko University of Kiev
*
∗ Postal address: Department of Mathematics, Utrecht University, Postbus 80010, 3508 TA Utrecht, The Netherlands. Email address: [email protected]
∗∗ Postal address: Faculty of Cybernetics, National Taras Shevchenko University of Kiev, 01033 Kiev, Ukraine.
∗∗ Postal address: Faculty of Cybernetics, National Taras Shevchenko University of Kiev, 01033 Kiev, Ukraine.
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Abstract

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We consider the Λ-coalescent processes with a positive frequency of singleton clusters. The class in focus covers, for instance, the beta(a, b)-coalescents with a > 1. We show that some large-sample properties of these processes can be derived by coupling the coalescent with an increasing Lévy process (subordinator), and by exploiting parallels with the theory of regenerative composition structures. In particular, we discuss the limit distributions of the absorption time and the number of collisions.

Keywords

Absorption timecouplingΛ-coalescentnumber of collisionsregenerative composition structuresubordinator

MSC classification

Secondary: 60C05: Combinatorial probability 60G09: Exchangeability 60F05: Central limit and other weak theorems 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K05: Renewal theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 4 , December 2011 , pp. 1133 - 1151
Copyright
Copyright © Applied Probability Trust 2011 

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