Random recursive trees and the bolthausen-sznitman coalescent

Electronic Journal of Probability

Volumn 10, Issue , 2005, Pages 718-745

  Goldschmidt, Christina ^a   Martin, James B ^b  

^a PEMBROKE COLLEGE   (United Kingdom)

^b UNIVERSITÉ PARIS DESCARTES   (France)

Author keywords

Coalescent; Partition; Random recursive tree

Indexed keywords

EID: 23244443030     PISSN: None     EISSN: 10836489     Source Type: Journal    
DOI: 10.1214/EJP.v10-265     Document Type: Article

References (28)

1. Richard Arratia, A. D. Barbour, and Simon Tavare. Logarithmic Combinatorial Structures: a Probabilistic Approach. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, 2003.
2. Anne-Laure Basdevant. Ruelle’s probability cascades seen as a fragmentation process. Preprint available at http://arxiv.org/abs/math.PR/0501088, 2005.
3. Jean Bertoin. Self-similar fragmentations. Ann. Inst. H. Poincare Probab. Statist., 38(3):319-340, 2002.
4. Jean Bertoin and Jean-François Le Gall. The Bolthausen-Sznitman coalescent and the genealogy of continuous-state branching processes. Probab. Theory Related Fields, 117(2):249-266, 2000.
5. E. Bolthausen and A.-S. Sznitman. On Ruelle’s probability cascades and an abstract cavity method. Comm. Math. Phys., 197(2):247-276, 1998.
6. J. M. DeLaurentis and B. G. Pittel. Random permutations and Brownian motion. Pacific J. Math., 119(2):287-301, 1985.
7. Peter Donnelly, Thomas G. Kurtz, and Simon Tavare. On the functional central limit theorem for the Ewens sampling formula. Ann. Appl. Probab., 1(4):539-545, 1991.
8. Shui Feng and Fred M. Hoppe. Large deviation principles for some random combinatorial structures in population genetics and Brownian motion. Ann. Appl. Probab., 8(4):975-994, 1998.
9. James Allen Fill, Nevin Kapur, and Alois Panholzer. Destruction of very simple trees. Preprint available at http://arxiv.org/abs/math.PR/0412155, 2004.
10. Jennie C. Hansen. A functional central limit theorem for the Ewens sampling formula. J. Appl. Probab., 27(1):28-43, 1990.
11. Svante Janson. Random cutting and records in deterministic and random trees. Tech. Report 2004:25, Department of Mathematics, Uppsala University; available from http://www.math.uu.se/∼svante/papers/index.html, 2004.
12. Svante Janson. Random records and cuttings in complete binary trees. In Mathematics and Computer Science III (Vienna 2004), Trends Math., pages 241-253. Birkhauser, Basel, 2004.
13. J. F. C. Kingman. The representation of partition structures. J. London Math. Soc. (2), 18(2):374-380, 1978.
14. J. F. C. Kingman. The coalescent. Stochastic Process. Appl., 13(3):235-248, 1982.
15. Philippe Marchal. Nested regenerative sets and their associated fragmentation process. In Mathematics and Computer Science III (Vienna 2004), Trends Math., pages 461-470. Birkhauser, Basel, 2004.
16. A. Meir and J. W. Moon. Cutting down random trees. J. Austral. Math. Soc., 11:313-324, 1970.
17. A. Meir and J. W. Moon. Cutting down recursive trees. Math. Bioscience, 21:173-181, 1974.
18. Alois Panholzer. Non-crossing trees revisited: cutting down and spanning subtrees. In Discrete random walks (Paris 2003), Discrete Math. Theor. Comput. Sci. Proc., AC, pages 265-276 (electronic). Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2003.
19. Alois Panholzer. Destruction of recursive trees. In Mathematics and Computer Science III (Vienna 2004), Trends Math., pages 267-280. Birkhauser, Basel, 2004.
20. N. Pippenger. An infinite product for e. Amer. Math. Monthly, 87(5):391, 1980.
21. Jim Pitman. Coalescents with multiple collisions. Ann. Probab., 27(4):1870-1902, 1999.
22. Jim Pitman. Combinatorial stochastic processes. To appear in Lectures on Probability Theory and Statistics (Saint-Flour). Springer-Verlag. Available at http://stat.berkeley.edu/users/pitman/621.pdf, 2002.
23. Jason Schweinsberg. A necessary and sufficient condition for the A-coalescent to come down from infinity. Electron. Comm. Probab., 5:1-11 (electronic), 2000.
24. Robert T. Smythe and Hosam M. Mahmoud. A survey of recursive trees. Teor. Imovir. Mat. Stat. (Theor. Probability and Math. Statist.), 51:1-29, 1994.
25. J. Sondow. A faster product for n and a new integral for ln n. To appear in Amer. Math. Monthly; available at http://arxiv.org/abs/math.NT/0401406, 2004.
26. Richard P. Stanley. Enumerative Combinatorics. Vol. 1, volume 49 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1997. With a foreword by Gian-Carlo Rota, Corrected reprint of the 1986 original.
27. J. van de Lune. An Introduction to Tauberian Theory: From Tauber to Wiener, volume 12 of CWI Syllabi. Stichting Mathematisch Centrum Centrum voor Wiskunde en Informatica, Amsterdam, 1986.
28. Herbert S. Wilf. generatingfunctionology. Academic Press Inc., Boston, MA, second edition, 1994.