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2020 Metrics on sets of interval partitions with diversity
Noah Forman, Soumik Pal, Douglas Rizzolo, Matthias Winkel
Electron. Commun. Probab. 25: 1-16 (2020). DOI: 10.1214/20-ECP317


We first consider interval partitions whose complements are Lebesgue-null and introduce a complete metric that induces the same topology as the Hausdorff distance (between complements). This is done using correspondences between intervals. Further restricting to interval partitions with $\alpha $-diversity, we then adjust the metric to incorporate diversities. We show that this second metric space is Lusin. An important feature of this topology is that path-continuity in this topology implies the continuous evolution of diversities. This is important in related work on tree-valued stochastic processes where diversities are branch lengths.


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Noah Forman. Soumik Pal. Douglas Rizzolo. Matthias Winkel. "Metrics on sets of interval partitions with diversity." Electron. Commun. Probab. 25 1 - 16, 2020.


Received: 5 July 2019; Accepted: 10 May 2020; Published: 2020
First available in Project Euclid: 9 June 2020

zbMATH: 07225531
MathSciNet: MR4112769
Digital Object Identifier: 10.1214/20-ECP317

Primary: 60J25 , 60J60 , 60J80
Secondary: 60G18 , 60G52 , 60G55

Keywords: $\alpha $-diversity , interval partition , Poisson–Dirichlet distribution


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