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A Construction of a β-Coalescent via the Pruning of Binary Trees

Part of: Markov processes

Published online by Cambridge University Press:  30 January 2018

Romain Abraham  and
Jean-François Delmas
Romain Abraham*
Affiliation:
Université d'Orléans
Jean-François Delmas*
Affiliation:
École des Ponts et Chaussées
*
∗ Postal address: Laboratoire MAPMO, CNRS, UMR 6628, Fédération Denis Poisson, FR 2964, Université d'Orléans, B.P. 6759, 45067 Orléans cedex 2, France. Email address: [email protected]
∗∗ Postal address: Université Paris-Est, École des Ponts, CERMICS, 6-8 Avenue Blaise Pascal, Champs-sur-Marne, 77455 Marne La Vallée, France.
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Abstract

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Considering a random binary tree with n labelled leaves, we use a pruning procedure on this tree in order to construct a β(3/2,1/2)-coalescent process. We also use the continuous analogue of this construction, i.e. a pruning procedure on Aldous's continuum random tree, to construct a continuous state space process that has the same structure as the β-coalescent process up to some time change. These two constructions enable us to obtain results on the coalescent process, such as the asymptotics on the number of coalescent events or the law of the blocks involved in the last coalescent event.

Keywords

Coalescent processbinary treepruningcontinuum random tree

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 3 , September 2013 , pp. 772 - 790
Copyright
© Applied Probability Trust 

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