Asymptotic results for coalescent processes without proper frequencies and applications to the two-parameter Poisson - Dirichlet coalescent

Stochastic Processes and their Applications

Volumn 120, Issue 11, 2010, Pages 2159-2173

  Mohle M ^a  

^a UNIVERSITY OF TÜBINGEN   (Germany)

Author keywords

Coalescent; External branch length; Recursion with random indices; Simultaneous multiple collisions; Total branch length; Two parameter Poisson Dirichlet coalescent

Indexed keywords

BRANCH LENGTH; COALESCENT; DIRICHLET; MULTIPLE COLLISIONS; RECURSIONS;

METHOD OF MOMENTS; RANDOM VARIABLES;

ASYMPTOTIC ANALYSIS;

EID: 77956395446     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.spa.2010.07.004     Document Type: Article

References (31)

1. A.-L. Basdevant, and C. Goldschmidt Asymptotics of the allele frequency spectrum associated with the BolthausenSznitman coalescent Electron. J. Probab. 13 2008 486 512 MR2386740
2. J. Berestycki, N. Berestycki, and J. Schweinsberg Beta-coalescents and continuous stable random tress Ann. Probab. 35 2007 1835 1887 MR2349577
3. J. Berestycki, N. Berestycki, and J. Schweinsberg Small-time behavior of Beta-coalescents Ann. Inst. H. Poincar Probab. Statist. 44 2008 214 238 MR2446321
4. J. Bertoin Two-parameter PoissonDirichlet measures and reversible exchangeable fragmentationcoalescence processes Combin. Probab. Comput. 17 2008 329 337 MR2410390
5. P. Billingsley Probability and Measure 3rd edition 1995 Wiley New York MR1324786
6. A. Caliebe, R. Neininger, M. Krawczak, and U. Rsler On the length distribution of external branches in coalescence trees: genetic diversity within species Theor. Popul. Biol. 72 2007 245 252 Zbl 1123.92024
7. P. Carmona, F. Petit, and M. Yor On the distribution and asymptotic results for exponential integrals of Lévy processes M. Yor, Exponential Functionals and Principal Values Related to Brownian Motion 1997 Biblioteca de la Revista Matematica Iberoamericana, Madrid 73 121 MR1648657
8. J.-F. Delmas, J.-S. Dhersin, and A. Siri-Jegousse Asymptotic results on the length of coalescent trees Ann. Appl. Probab. 18 2008 997 1025 MR2418236
9. M. Drmota, A. Iksanov, M. Möhle, and U. Rsler Asymptotic results concerning the total branch length of the BolthausenSznitman coalescent Stochastic Process. Appl. 117 2007 1404 1421 MR2353033
10. M. Drmota, A. Iksanov, M. Möhle, and U. Rsler A limiting distribution for the number of cuts needed to isolate the root of a random recursive tree Random Struct. Algorithms 34 2009 319 336 MR2504401
11. F. Freund, Almost sure asymptotics for mutated external branches of coalescent processes and applications to the number of types, Preprint, 2009.
12. F. Freund, and M. Möhle On the number of allelic types for samples taken from exchangeable coalescents with mutation Adv. Appl. Probab. 41 2009 1082 1101
13. F. Freund, and M. Möhle On the time back to the most recent common ancestor and the external branch length of the BolthausenSznitman coalescent Markov Process. Related. Fields 15 2009 387 416 MR2554368
14. A. Gnedin, B. Hansen, and J. Pitman Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws Probab. Surv. 4 2007 146 171 MR2318403
15. A. Gnedin, A. Iksanov, and M. Möhle On asymptotics of exchangeable coalescents with multiple collisions J. Appl. Probab. 45 2008 1186 1195 MR2484170
16. B. Haas, G. Miermont, Self-similar scaling limits of non-increasing Markov chains, Preprint, 2009.
17. K. Handa The two-parameter PoissonDirichlet point process Bernoulli 15 2009 1082 1116 MR2597584
18. L.F. James, Poisson Dirichlet(α, θ)-bridge equations and coagulationfragmentation duality, Preprint, 2009.
19. G. Kersting, The external length in Kingman's coalescent, Preprint, 2010.
20. J.F.C. Kingman On the genealogy of large populations J. Appl. Probab. 19A 1982 27 43 MR0633178
21. J.F.C. Kingman The coalescent Stochastic Process. Appl. 13 1982 235 248 MR0671034
22. J.F.C. Kingman The representation of partition structures J. Lond. Math. Soc. (2) 18 1978 374 380 MR0509954
23. A. Marynych, On the asymptotics of moments of linear random recurrences, Preprint, 2009.
24. M. Möhle On the number of segregating sites for populations with large family sizes Adv. Appl. Probab. 38 2006 750 767 MR2256876
25. M. Möhle On sampling distributions for coalescent processes with simultaneous multiple collisions Bernoulli 12 2006 35 53 MR2202319
26. M. Möhle, and S. Sagitov A classification of coalescent processes for haploid exchangeable population models Ann. Probab. 29 2001 1547 1562 MR2024501
27. J. Pitman Coalescents with multiple collisions Ann. Probab. 27 1999 1870 1902 MR1742892
28. J. Pitman Combinatorial Stochastic Processes Lecture Notes in Mathematics vol. 1875 2006 Springer New York MR2245368
29. J. Pitman, and M. Yor The two-parameter PoissonDirichlet distribution derived from a stable subordinator Ann. Probab. 25 1997 855 900 MR1434129
30. S. Sagitov Convergence to the coalescent with simultaneous multiple mergers J. Appl. Probab. 40 2003 839 854 MR2012671
31. J. Schweinsberg Coalescents with simultaneous multiple collisions Electron. J. Probab. 5 2000 1 50 MR1781024