Small-time behavior of beta coalescents

Annales de l'institut Henri Poincare (B) Probability and Statistics

Volumn 44, Issue 2, 2008, Pages 214-238

  Berestycki, Julien ^a   Berestycki, Nathanaél ^b   Schweinsberg, Jason ^c  

^a AIX MARSEILLE UNIVERSITÉ   (France)

^b UNIVERSITY OF BRITISH COLUMBIA   (Canada)

^c Department of Mathematics 0112 ^*  (United States)

Author keywords

Coalescence; Coalescent with multiple mergers; Continuous state branching process

Indexed keywords

EID: 58449119509     PISSN: 02460203     EISSN: 02460203     Source Type: Journal    
DOI: 10.1214/07-AIHP103     Document Type: Article

References (36)

1. D. J. Aldous. Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5 (1999) 3-48. MR1673235
2. A.-L. Basdevant. Ruelle's probability cascades seen as a fragmentation process. Markov Process. Related Fields 12 (2006) 447-474. MR2246260
3. J. Berestycki, N. Berestycki and J. Schweinsberg. Beta-coalescents and continuous stable random trees. Ann. Probab. 35 (2007) 1835-1887.
4. J. Bertoin. Lévy Processes. Cambridge University Press, Cambridge, 1996. MR1406564
5. J. Bertoin. Random Coagulation and Fragmentation Processes. Cambridge University Press, Cambridge, 2006. MR2253162
6. J. Bertoin and J.-F. Le Gall. The Bolthausen-Sznitman coalescent and the genealogy of continuous-state branching processes. Probab. Theory Related Fields 117 (2000) 249-266. MR1771663
7. J. Bertoin and J.-F. Le Gall. Stochastic flows associated to coalescent processes. Probab. Theory Related Fields 126 (2003) 261-288. MR1990057
8. J. Bertoin and J.-F. Le Gall. Stochastic flows associated to coalescent processes III: limit theorems. Illinois J. Math. 50 (2006) 147-181. MR2247827
9. J. Bertoin and J. Pitman. Two coalescents derived from the ranges of stable subordinators. Electron. J. Probab. 5 (2000) 1-17. MR1768841
10. N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Cambridge University Press, Cambridge, 1987. MR0898871
11. M. Birkner, J. Blath, M. Capaldo, A. Etheridge,M.Möhle, J. Schweinsberg and A.Wakolbinger. Alpha-stable branching and beta-coalescents. Electron. J. Probab. 10 (2005) 303-325. MR2120246
12. E. Bolthausen and A.-S. Sznitman. On Ruelle's probability cascades and an abstract cavity method. Comm. Math. Phys. 197 (1998) 247-276. MR1652734
13. P. Donnelly, S. N. Evans, K. Fleischmann, T. G. Kurtz and X. Zhou. Continuum-sites stepping-stone models, coalescing exchangeable partitions, and random trees. Ann. Probab. 28 (2000) 1063-1110. MR1797304
14. P. Donnelly and T. G. Kurtz. Particle representations for measure-valued population models. Ann. Probab. 27 (1999) 166-205. MR1681126
15. R. M. Dudley. Real Analysis and Probability. Wadsworth and Brooks/Cole, Pacific Grove, CA, 1989. MR0982264
16. T. Duquesne. A limit theorem for the contour process of conditioned Galton-Watson trees. Ann. Probab. 31 (2003) 996-1027. MR1964956
17. T. Duquesne and J.-F. Le Gall. Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 (2005) 553-603. MR2147221
18. R. Durrett and J. Schweinsberg. A coalescent model for the effect of advantageous mutations on the genealogy of a population. Stochastic Process. Appl. 115 (2005) 1628-1657. MR2165337
19. N. El Karoui and S. Roelly. Propriétés de martingales, explosion et représentation de Lévy-Khintchine d'une classe de processus de branchement á valeurs mesures. Stochastic Process. Appl. 38 (1991) 239-266. MR1119983
20. S. N. Evans. Kingman's coalescent as a random metric space. In Stochastic Models: A Conference in Honour of Professor Donald A. Dawson (L. G. Gorostiza and B. G. Ivanoff, Eds). Canadian Mathematical Society/American Mathematical Society, 2000. MR1775475
21. K. Falconer. Fractal Geometry: Mathematical Foundations and Applications, 2nd edition. Wiley, Hoboken, NJ, 2003. MR2118797
22. C. Goldschmidt and J. Martin. Random recursive trees and the Bolthausen-Sznitman coalescent. Electron. J. Probab. 10 (2005) 718-745. MR2164028
23. J. Kesten, P. Ney and F. Spitzer. The Galton-Watson process with mean one and finite variance. Theory Probab. Appl. 11 (1966) 513-540. MR0207052
24. J. F. C. Kingman. The representation of partition structures. J. London Math. Soc. 18 (1978) 374-380. MR0509954
25. J. F. C. Kingman. The coalescent. Stochastic Process. Appl. 13 (1982) 235-248. MR0671034
26. J. Lamperti. The limit of a sequence of branching processes. Z. Wahrsch. Verw. Gebiete 7 (1967) 271-288. MR0217893
27. J. Lamperti. Continuous state branching processes. Bull. Amer. Math. Soc. 73 (1967) 382-386. MR0208685
28. M. Möhle. On the number of segregating sites for populations with large family sizes. Adv. in Appl. Probab. 38 (2006) 750-767.
29. M. Möhle and S. Sagitov. A classification of coalescent processes for haploid exchangeable population models. Ann. Probab. 29 (2001) 1547-1562. MR1880231 (Pubitemid 33406000)
30. J. Pitman. Coalescents with multiple collisions. Ann. Probab. 27 (1999) 1870-1902. MR1742892
31. S. Sagitov. The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 (1999) 1116-1125. MR1742154
32. J. Schweinsberg. A necessary and sufficient condition for the λ-coalescent to come down from infinity. Electron. Comm. Probab. 5 (2000) 1-11. MR1736720
33. J. Schweinsberg. Coalescent processes obtained from supercritical Galton-Watson processes. Stochastic Process. Appl. 106 (2003) 107-139. MR1983046
34. M. L. Silverstein. A new approach to local times. J. Math. Mech. 17 (1968) 1023-1054. MR0226734
35. R. Slack. A branching process with mean one and possibly infinite variance. Z. Wahrsch. Verw. Gebiete 9 (1968) 139-145. MR0228077
36. A. M. Yaglom. Certain limit theorems of the theory of branching processes. Dokl. Acad. Nauk SSSR 56 (1947) 795-798. MR0022045