A construction of a β-coalescent via the pruning of binary trees

Journal of Applied Probability

Volumn 50, Issue 3, 2013, Pages 772-790

  Abraham, Romain ^a   Delmas, Jean François ^b  

^a UNIVERSITÉ D'ORLÉANS   (France)

^b UNIVERSITÉ PARIS EST   (France)

Author keywords

Binary tree; Coalescent process; Continuum random tree; Pruning

Indexed keywords

EID: 84885104344     PISSN: 00219002     EISSN: None     Source Type: Journal    
DOI: 10.1239/jap/1378401235     Document Type: Article

References (25)

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