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Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees)
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  • Published: 14 August 2008

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

  • Andreas Greven1,
  • Peter Pfaffelhuber2 &
  • Anita Winter1 

Probability Theory and Related Fields volume 145, pages 285–322 (2009)Cite this article

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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.

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Author information

Authors and Affiliations

  1. Mathematisches Institut, University of Erlangen, Bismarckstr. 1½, 91054, Erlangen, Germany

    Andreas Greven & Anita Winter

  2. Fakultät für Mathematik und Physik, Albert-Ludwigs-Universität Freiburg, Eckerstr. 1, 79104, Freiburg, Germany

    Peter Pfaffelhuber

Authors
  1. Andreas Greven
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  2. Peter Pfaffelhuber
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  3. Anita Winter
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Corresponding author

Correspondence to Anita Winter.

Additional information

The research was supported by the DFG-Forschergruppe 498 via grant GR 876/13-1,2.

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Cite this article

Greven, A., Pfaffelhuber, P. & Winter, A. Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees). Probab. Theory Relat. Fields 145, 285–322 (2009). https://doi.org/10.1007/s00440-008-0169-3

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  • Received: 28 September 2006

  • Revised: 02 June 2008

  • Published: 14 August 2008

  • Issue Date: September 2009

  • DOI: https://doi.org/10.1007/s00440-008-0169-3

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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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