On Λ-coalescents with dust component

Journal of Applied Probability

Volumn 48, Issue 4, 2011, Pages 1133-1151

  Gnedin, Alexander ^a   Iksanov, Alexander ^b   Marynych, Alexander ^b  

^a UTRECHT UNIVERSITY   (Netherlands)

^b TARAS SHEVCHENKO NATIONAL UNIVERSITY OF KYIV   (Ukraine)

Author keywords

coalescent; Absorption time; Coupling; Number of collisions; Regenerative composition structure; Subordinator

Indexed keywords

EID: 84855294897     PISSN: 00219002     EISSN: None     Source Type: Journal    
DOI: 10.1239/jap/1324046023     Document Type: Article

References (27)

1. Barbour, A. D. and Gnedin, A. V. (2006). Regenerative compositions in the case of slow variation. Stoch. Process. Appl. 116, 1012-1047. (Pubitemid 43949062)
2. Berestycki, N. (2009). Recent Progress in Coalescent Theory (Ensaios Matemáticos 16). Sociedade Brasileira de Matemática, Rio de Janeiro.
3. Bertoin, J. (1996). Subordinators: Examples and Applications (Lecture Notes Math. 1727). Springer, Berlin.
4. Bertoin, J. (2010). Exchangeable coalescents. Lecture Notes, ETH Zürich. Available at http://www.fim.math.ethz.ch/lectures/Lectures-Bertoin. pdf.
5. Bingham, N. H. (1972). Limit theorems for regenerative phenomena, recurrent events and renewal theory. Z. Wahrscheinlichkeitsth. 21, 20-44.
6. Bingham N. H., Goldie C. M. and Teugels, J. L. (1989). Regular Variation. Cambridge University Press.
7. Drmota, M., Iksanov, A., Moehle, M. and Roesler, U. (2009). A limiting distribution for the number of cuts needed to isolate the root of a random recursive tree. Random Structures Algorithms 34, 319-336.
8. Durrett,R. and Liggett,T.M. (1983). Fixed points of the smoothing transformation. Z.Wahrscheinlichkeitsth. 64, 275-301.
9. Freund, F. and Möhle, M. (2009). On the time back to the most recent common ancestor and the external branch length of the Bolthausen-Sznitman coalescent. Markov Process. Relat. Fields 15, 387-416.
10. Gnedin, A. and Pitman, J. (2005). Regenerative composition structures. Ann. Prob. 33, 445-479.
11. Gnedin, A. and Yakubovich, Y. (2007). On the number of collisions in Λ-coalescents. Electron. J. Prob. 12, 1547-1567.
12. Gnedin, A., Iksanov, A. and Marynych, A. (2010). Limit theorems for the number of occupied boxes in the Bernoulli sieve. Theory Stoch. Process. 16, 44-57.
13. Gnedin, A., Iksanov, A. and Möhle, M. (2008). On asymptotics of exchangeable coalescents with multiple collisions. J. Appl. Prob. 45, 1186-1195.
14. Gnedin, A., Iksanov, A. and Roesler, U. (2008). Small parts in the Bernoulli sieve. Discrete Math. Theoret. Comput. Sci. AI, 235-242.
15. Gnedin, A., Pitman, J. and Yor, M. (2006). Asymptotic laws for compositions derived from transformed subordinators. Ann. Prob. 34, 468-492. (Pubitemid 43875114)
16. Gnedin, A., Pitman, J. and Yor, M. (2006). Asymptotic laws for regenerative compositions: gamma subordinators and the like. Prob. Theory Relat. Fields 135, 576-602. (Pubitemid 43746109)
17. Goldschmidt, C. and Martin, J. B. (2005). Random recursive trees and the Bolthausen-Sznitman coalescent. Electron. J. Prob. 10, 718-745.
18. Haas, B. and Miermont, G. (2011). Self-similar scaling limits of non-increasing Markov chains. To appear in Bernoulli.
19. Iksanov, A. and Möhle, M. (2007). A probabilistic proof of a weak limit law for the number of cuts needed to isolate the root of a random recursive tree. Electron. Commun. Prob. 12, 28-35.
20. Iksanov, A. and Möhle, M. (2008). On the number of jumps of random walks with a barrier. Adv. Appl. Prob. 40, 206-228. (Pubitemid 351772465)
21. Iksanov, A., Marynych, A. and Möhle, M. (2009). On the number of collisions in beta(2, b)-coalescents. Bernoulli 15, 829-845.
22. Möhle, M. (2010). Asymptotic results for coalescent processes without proper frequencies and applications to the two-parameter Poisson-Dirichlet coalescent. Stoch. Process. Appl. 120, 2159-2173
23. Negadailov, P. (2010). Limit theorems for random recurrences and renewal-type processes. Doctoral Thesis, Utrecht University. Available at http://igitur-archive.library.uu.nl/dissertations/.
24. Pitman, J. (1999). Coalescents with multiple collisions. Ann. Prob. 27, 1870-1902.
25. Pitman, J. (2006). Combinatorial Stochastic Processes (Lecture Notes Math. 1875). Springer, Berlin.
26. Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Prob. 36, 1116-1125.
27. Schweinsberg, J. (2000).A necessary and sufficient condition for the-coalescent to come down from infinity. Electron. Commun. Prob. 5, 1-11.