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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 

References

Abraham, R. and Delmas, J.-F. (2012). A continuum tree-valued Markov process. Ann. Prob. 40, 1167–1211.CrossRefGoogle Scholar
Abraham, R. and Delmas, J.-F. (2013). Record process on the continuum random tree. Latin Amer. J. Prob. Math. Statist. 10, 225–251.Google Scholar
Abraham, R. and Serlet, L. (2002). Poisson snake and fragmentation. Electron. J. Prob. 7, 1–15.Google Scholar
Abraham, R., Delmas, J.-F. and He, H. (2013). Pruning of CRT sub-trees. Submitted. Available at http://arxiv.org/abs/1212.2765v1.Google Scholar
Abraham, R., Delmas, J.-F. and Voisin, G. (2010). Pruning a Lévy continuum random tree. Electron. J. Prob. 15, 1429–1473.Google Scholar
Aldous, D. (1991). The continuum random tree. I. Ann. Prob. 19, 1–28.CrossRefGoogle Scholar
Aldous, D. (1993). The continuum random tree. III. Ann. Prob. 21, 248–289.Google Scholar
Aldous, D. and Pitman, J. (1998). The standard additive coalescent. Ann. Prob. 26, 1703–1726.Google Scholar
Berestycki, N. (2009). Recent Progress in Coalescent Theory (Ensaios Matemáticos 16). Sociedade Brasileira de Matemática, Rio de Janeiro.Google Scholar
Berestycki, J., Berestycki, N. and Schweinsberg, J. (2007). Beta-coalescents and continuous stable random trees. Ann. Prob. 35, 1835–1887.CrossRefGoogle Scholar
Bertoin, J. and Yor, M. (2005). Exponential functionals of Lévy processes. Prob. Surveys 2, 191–212.Google Scholar
Birkner, M. et al. (2005). {Alpha-stable branching and beta-coalescents.} Electron. J. Prob. 10, 303–325.Google Scholar
Bolthausen, E. and Sznitman, A.-S. (1998). On Ruelle's probability cascades and an abstract cavity method. Commun. Math. Phys. 197, 247–276.Google Scholar
Delmas, J.-F. and Chen, Y.-T. (2012). Smaller population size at the MRCA time for stationary branching processes. Ann. Prob. 40, 2034–2068.Google Scholar
Duquesne, T. and Le Gall, J.-F. (2002). Random Trees, Lévy Processes and Spatial Branching Processes (Astérisque 281). Societe Mathematique de France.Google Scholar
Foucard, C. and Hénard, O. (2013). Stable continuous branching processes with immigration and Beta-Fleming-Viot processes with immigration. Electron. J. Prob. 18, 1–21.Google Scholar
Goldschmidt, C. and Martin, J. (2005). Random recursive trees and the Bolthausen-Sznitman coalescent. Electron. J. Prob. 10, 718–745.Google Scholar
Gnedin, A., Iksanov, A. and Marynych, A. (2011). On lambda-coalescents with dust component. J. Appl. Prob. 48, 1133–1151.Google Scholar
Hoscheit, P. (2013). Fluctuations for the number of records on random binary trees. Submitted. Available at http://arxiv.org/abs/1212.5434v1.Google Scholar
Iksanov, A. and Möhle, M. (2008). On the number of Jumps of random walks with a barrier. Adv. Appl. Prob. 40, 206–228.Google Scholar
Janson, S. (2006). Random cutting and records in deterministic and random trees. Random Structures Algorithms 29, 139–179.CrossRefGoogle Scholar
Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248.Google Scholar
Le Gall, J.-F. (2006). Random real trees. Ann. Fac. Sci. Toulouse Math. 15, 35–62.Google Scholar
Pitman, J. (1999). Coalescents with multiple collisions. Ann. Prob. 27, 1870–1902.CrossRefGoogle Scholar
Pitman, J. (2006). Combinatorial Stochastic Processes (Lectures Notes Math. 1875). Springer, Berlin.Google Scholar