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)

Article

Abstract

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.

Keywords

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 

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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

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