Convergence in distribution of random metric measure spaces (The Lambda-coalescent measure tree)

Pfaffelhuber P, Winter A, Greven A (2009)


Publication Type: Journal article

Publication year: 2009

Journal

Publisher: Springer Verlag (Germany)

Book Volume: 145

Pages Range: 285-322

Journal Issue: 1

URI: https://link.springer.com/article/10.1007/s00440-008-0169-3

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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How to cite

APA:

Pfaffelhuber, P., Winter, A., & Greven, A. (2009). Convergence in distribution of random metric measure spaces (The Lambda-coalescent measure tree). Probability Theory and Related Fields, 145(1), 285-322.

MLA:

Pfaffelhuber, Peter, Anita Winter, and Andreas Greven. "Convergence in distribution of random metric measure spaces (The Lambda-coalescent measure tree)." Probability Theory and Related Fields 145.1 (2009): 285-322.

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