Convergence in distribution of random metric measure spaces: (Lambda-coalescent measure trees)
Convergence in distribution of random metric measure spaces: (Lambda-coalescent measure trees)
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随机度量测度空间分布的收敛性：（Lambda 合并测度树）
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
A. Winter
中科院分区：
文献类型：
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作者：
A. Greven;P. Pfaffelhuber;A. Winter
我们考虑赋以概率度量的完备可分度量空间。基于度量度量空间序列收敛的思想，给出了收敛的概念，当且仅当从这些空间采样的所有有限子空间收敛。这种拓扑是按照Gromov的思想将两个度量空间等距嵌入到一个公共度量空间中，并结合固定度量空间上的概率度量之间的Prohorov度量。我们证明了对于这种拓扑，只要序列是紧的，分布上的收敛就是所有随机抽样的有限子空间的收敛。我们基于容易计算的量给出了紧性的一个特征。特别感兴趣的子空间是实树空间和配备概率度量的超度量空间。作为一个例子，我们刻画了由Lambda-Coaldium的随机谱系给出的(超)度量空间在分布上的收敛。我们证明了当且仅当所谓的“无尘”性质成立时，Lambda-联合定义了一个无限(随机)度量空间。
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.