Probability Theory and Related Fields

, Volume 145, Issue 1–2, pp 285–322 | Cite as

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



We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge. This topology is metrized following Gromov’s idea of embedding two metric spaces isometrically into a common metric space combined with the Prohorov metric between probability measures on a fixed metric space. We show that for this topology convergence in distribution follows—provided the sequence is tight—from convergence of all randomly sampled finite subspaces. We give a characterization of tightness based on quantities which are reasonably easy to calculate. Subspaces of particular interest are the space of real trees and of ultra-metric spaces equipped with a probability measure. As an example we characterize convergence in distribution for the (ultra-)metric measure spaces given by the random genealogies of the Λ-coalescents. We show that the Λ-coalescent defines an infinite (random) metric measure space if and only if the so-called “dust-free”-property holds.


Metric measure spaces Gromov metric triple \({\mathbb{R}}\) -trees Gromov–Hausdorff topology weak topology Prohorov metric Wasserstein metric Λ-Coalescent 

Mathematics Subject Classification (2000)

Primary: 60B10 05C80 Secondary: 60B05 60G09 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aldous D.: The random walk construction of uniform spanning trees and uniform labeled trees. SIAM J. Discr. Math 3, 450–465 (1990)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Aldous D.: The continuum random tree III. Ann. Prob. 21, 248–289 (1993)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Birkner M., Blath J., Capaldo M., Etheridge A., Möhle M., Schweinsberg J., Wakolbinger A.: Alpha-stable branching and beta coalescents. Elec. J. Prob. 10, 303–325 (2005)Google Scholar
  4. 4.
    Burago, D., Burago, Y., Ivanov, S.: A course in metric geometry, graduate studies in mathematics, vol. 33. AMS, Providence (2001)Google Scholar
  5. 5.
    Berestycki J., Berestycki N., Schweinsberg J.: Small-time behavior of beta coalescents. Ann. Inst. H. Poin. 44(2), 214–238 (2008)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Bertoin J., Le Gall J.-F.: Stochastic flows associated to coalescent processes III: Limit theorems. Illinois J. Math. 50(14), 147–181 (2006)MATHMathSciNetGoogle Scholar
  7. 7.
    Bridson M.R., Haefliger A.: Metric spaces of non-positive curvature. Springer, Heidelberg (1999)MATHGoogle Scholar
  8. 8.
    Bolthausen E., Kistler N.: On a non-hierarchical model of the generalized random energy model. Ann. Appl. Prob. 16(1), 1–14 (2006)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Drmota M., Hwuang H.-K.: Profiles of random trees: correlation and width of random recursive trees and binary search trees. Adv. Appl. Prob. 37(2), 321–341 (2005)MATHCrossRefGoogle Scholar
  10. 10.
    Ethier S.N., Kurtz T.: Markov Processes. Characterization and Convergence. Wiley, New York (1986)MATHGoogle Scholar
  11. 11.
    Evans S.N., Pitman J., Winter A.: Rayleigh processes, real trees, and root growth with re-grafting. Prob. Theo. Rel. Fields 134(1), 81–126 (2006)MATHCrossRefMathSciNetGoogle Scholar
  12. 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)Google Scholar
  13. 13.
    Evans S.N., Winter A.: Subtree prune and re-graft: A reversible real-tree valued Markov chain. Ann. Prob. 34(3), 918–961 (2006)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Felsenstein J.: Inferring Phylogenies. Sinauer, Sunderland (2003)Google Scholar
  15. 15.
    Greven, A., Limic, V., Winter, A.: Cluster formation in spatial Moran models in critical dimension via particle representation. Submitted (2007)Google Scholar
  16. 16.
    Greven, A., Pfaffelhuber, P., Winter, A.: Tree-valued resampling dynamics—martingale problems and applications. Submitted (2008)Google Scholar
  17. 17.
    Gromov M.: Metric structures for Riemannian and non-Riemannian spaces. Birkhäuser, Basel (1999)MATHGoogle Scholar
  18. 18.
    Gibbs A., Su F.: On choosing and bounding probability metrics. Intl. Stat. Rev. 7(3), 419–435 (2002)CrossRefGoogle Scholar
  19. 19.
    Hudson R.R.: Gene genealogies and the coalescent process. Oxford Surv Evol. Biol. 9, 1–44 (1990)Google Scholar
  20. 20.
    Kallenberg O.: Foundations of Modern Probability. Springer, Heidelberg (2002)MATHGoogle Scholar
  21. 21.
    Limic V., Sturm A.: The spatial Λ-coalescent. Elec. J. Prob. 11, 363–393 (2006)MathSciNetGoogle Scholar
  22. 22.
    Mezard, M., Parisi, G., Virasoro, M.A.: The spin glass theory and beyond. In: World Scientific Lecture Notes in Physics, vol. 9 (1987)Google Scholar
  23. 23.
    Möhle M., Sagitov S.: A classification of coalescent processes for haploid exchangeable population models. Ann. Prob. 29, 1547–1562 (2001)MATHCrossRefGoogle Scholar
  24. 24.
    Pitman J.: Coalescents with multiple collisions. Ann. Prob. 27(4), 1870–1902 (1999)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Rachev S.T.: Probability Metrics and the stability of stochastic models. Wiley, New York (1991)MATHGoogle Scholar
  26. 26.
    Sagitov S.: The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Prob. 36(4), 1116–1125 (1999)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Sturm K.-T.: On the geometry of metric measure spaces. Acta Math. 196(1), 65–131 (2006)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Vershik A.M.: The universal Urysohn space, Gromov metric triples and random matrices on the natural numbers. Russ. Math. Surv. 53(3), 921–938 (1998)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Andreas Greven
    • 1
  • Peter Pfaffelhuber
    • 2
  • Anita Winter
    • 1
  1. 1.Mathematisches InstitutUniversity of ErlangenErlangenGermany
  2. 2.Fakultät für Mathematik und PhysikAlbert-Ludwigs-Universität FreiburgFreiburgGermany

Personalised recommendations