Electronic Communications in Probability

Metrics on sets of interval partitions with diversity

Noah Forman, Soumik Pal, Douglas Rizzolo, and Matthias Winkel

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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.

Article information

Electron. Commun. Probab., Volume 25 (2020), paper no. 38, 16 pp.

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

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Primary: 60J25: Continuous-time Markov processes on general state spaces 60J60: Diffusion processes [See also 58J65] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G18: Self-similar processes 60G52: Stable processes 60G55: Point processes

interval partition Poisson–Dirichlet distribution $\alpha $-diversity

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Forman, Noah; Pal, Soumik; Rizzolo, Douglas; Winkel, Matthias. Metrics on sets of interval partitions with diversity. Electron. Commun. Probab. 25 (2020), paper no. 38, 16 pp. doi:10.1214/20-ECP317. https://projecteuclid.org/euclid.ecp/1591668057

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