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2020 Exchangeable hierarchies and mass-structure of weighted real trees
Noah Forman
Electron. J. Probab. 25: 1-28 (2020). DOI: 10.1214/20-EJP522

Abstract

Rooted, weighted continuum random trees are used to describe limits of sequences of random discrete trees. Formally, they are random quadruples $(\mathcal {T},d,r,p)$, where $(\mathcal {T},d)$ is a tree-like metric space, $r\in \mathcal {T}$ is a distinguished root, and $p$ is a probability measure on this space. Intuitively, these trees have a combinatorial “underlying branching structure” implied by their topology but otherwise independent of the metric $d$. We explore various ways of making this rigorous, using the weight $p$ to do so without losing the fractal complexity possible in continuum trees. We introduce a notion of mass-structural equivalence and show that two rooted, weighted $\mathbb {R}$-trees are equivalent in this sense if and only if the discrete hierarchies derived by i.i.d. sampling from their weights, in a manner analogous to Kingman’s paintbox, have the same distribution. We introduce a family of trees, called “interval partition trees” that serve as representatives of mass-structure equivalence classes, and which naturally represent the laws of the aforementioned hierarchies.

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Noah Forman. "Exchangeable hierarchies and mass-structure of weighted real trees." Electron. J. Probab. 25 1 - 28, 2020. https://doi.org/10.1214/20-EJP522

Information

Received: 17 March 2018; Accepted: 6 September 2020; Published: 2020
First available in Project Euclid: 28 October 2020

MathSciNet: MR4169172
Digital Object Identifier: 10.1214/20-EJP522

Subjects:
Primary: 60B05, 60C05, 60G09

JOURNAL ARTICLE
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Vol.25 • 2020
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