## Electronic Communications in Probability

### Metrics on sets of interval partitions with diversity

#### Abstract

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

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

Dates
Accepted: 10 May 2020
First available in Project Euclid: 9 June 2020

https://projecteuclid.org/euclid.ecp/1591668057

Digital Object Identifier
doi:10.1214/20-ECP317

#### Citation

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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