Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees)
成果类型：
Article
署名作者：
Greven, Andreas; Pfaffelhuber, Peter; Winter, Anita
署名单位：
University of Erlangen Nuremberg; University of Freiburg
刊物名称：
PROBABILITY THEORY AND RELATED FIELDS
ISSN/ISSBN：
0178-8051
DOI：
10.1007/s00440-008-0169-3
发表日期：
2009
页码：
285-322
关键词：
摘要：
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 Lambda-coalescents. We show that the Lambda-coalescent defines an infinite (random) metric measure space if and only if the so-called dust-free property holds.