Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees)

Probability Theory and Related Fields

Volumn 145, Issue 1-2, 2009, Pages 285-322

  Greven, Andreas ^a   Pfaffelhuber, Peter ^b   Winter, Anita ^a  

^a UNIVERSITY OF ERLANGEN NUREMBERG   (Germany)

^b UNIVERSITY OF FREIBURG   (Germany)

Author keywords

Coalescent; trees; Gromov metric triple; Gromov Hausdorff topology; Metric measure spaces; Prohorov metric; Wasserstein metric; Weak topology

Indexed keywords

EID: 67649092284     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00440-008-0169-3     Document Type: Article

References (28)

1. D. Aldous 1990 The random walk construction of uniform spanning trees and uniform labeled trees SIAM J. Discr. Math 3 450 465
2. D. Aldous 1993 The continuum random tree III Ann. Prob. 21 248 289
3. M. Birkner J. Blath M. Capaldo A. Etheridge M. Möhle J. Schweinsberg A. Wakolbinger 2005 Alpha-stable branching and beta coalescents Elec. J. Prob. 10 303 325
4. Burago, D., Burago, Y., Ivanov, S.: A course in metric geometry, graduate studies in mathematics, vol. 33. AMS, Providence (2001)
5. J. Berestycki N. Berestycki J. Schweinsberg 2008 Small-time behavior of beta coalescents Ann. Inst. H. Poin. 44 2 214 238
6. J. Bertoin J.-F. Le Gall 2006 Stochastic flows associated to coalescent processes III: Limit theorems Illinois J. Math. 50 14 147 181
7. Bridson M.R., Haefliger A.: Metric spaces of non-positive curvature. Springer, Heidelberg (1999)
8. E. Bolthausen N. Kistler 2006 On a non-hierarchical model of the generalized random energy model Ann. Appl. Prob. 16 1 1 14
9. M. Drmota H.-K. Hwuang 2005 Profiles of random trees: correlation and width of random recursive trees and binary search trees Adv. Appl. Prob. 37 2 321 341 (Pubitemid 40986390)
10. Ethier S.N., Kurtz T.: Markov Processes. Characterization and Convergence. Wiley, New York (1986)
11. S.N. Evans J. Pitman A. Winter 2006 Rayleigh processes, real trees, and root growth with re-grafting Prob. Theo. Rel. Fields 134 1 81 126 (Pubitemid 41697436)
12. Evans, S.N.: Kingman's coalescent as a random metric space. In: Gorostiza, L.G., Ivanoff, B.G. (eds.) Stochastic Models: Proceedings of the International Conference on Stochastic Models in Honour of Professor Donald A. Dawson, Ottawa, Canada, June 10-13, 1998. Canadian Mathematical Society, Ottawa (2000)
13. S.N. Evans A. Winter 2006 Subtree prune and re-graft: A reversible real-tree valued Markov chain Ann. Prob. 34 3 918 961 (Pubitemid 44091300)
14. Felsenstein J.: Inferring Phylogenies. Sinauer, Sunderland (2003)
15. Greven, A., Limic, V., Winter, A.: Cluster formation in spatial Moran models in critical dimension via particle representation. Submitted (2007)
16. Greven, A., Pfaffelhuber, P., Winter, A.: Tree-valued resampling dynamics-martingale problems and applications. Submitted (2008)
17. Gromov M.: Metric structures for Riemannian and non-Riemannian spaces. Birkhäuser, Basel (1999)
18. A. Gibbs F. Su 2002 On choosing and bounding probability metrics Intl. Stat. Rev. 7 3 419 435
19. R.R. Hudson 1990 Gene genealogies and the coalescent process Oxford Surv Evol. Biol. 9 1 44
20. Kallenberg O.: Foundations of Modern Probability. Springer, Heidelberg (2002)
21. V. Limic A. Sturm 2006 The spatial Λ-coalescent Elec. J. Prob. 11 363 393
22. Mezard, M., Parisi, G., Virasoro, M.A.: The spin glass theory and beyond. In: World Scientific Lecture Notes in Physics, vol. 9 (1987)
23. M. Möhle S. Sagitov 2001 A classification of coalescent processes for haploid exchangeable population models Ann. Prob. 29 1547 1562 (Pubitemid 33406000)
24. J. Pitman 1999 Coalescents with multiple collisions Ann. Prob. 27 4 1870 1902
25. Rachev S.T.: Probability Metrics and the stability of stochastic models. Wiley, New York (1991)
26. S. Sagitov 1999 The general coalescent with asynchronous mergers of ancestral lines J. Appl. Prob. 36 4 1116 1125
27. K.-T. Sturm 2006 On the geometry of metric measure spaces Acta Math. 196 1 65 131 (Pubitemid 44598869)
28. A.M. Vershik 1998 The universal Urysohn space, Gromov metric triples and random matrices on the natural numbers Russ. Math. Surv. 53 3 921 938