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October 2021 Exchangeable coalescents, ultrametric spaces, nested interval-partitions: A unifying approach
Félix Foutel-Rodier, Amaury Lambert, Emmanuel Schertzer
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Ann. Appl. Probab. 31(5): 2046-2090 (October 2021). DOI: 10.1214/20-AAP1641


Kingman’s (1978) representation theorem (J. Lond. Math. Soc. (2) 18 (1978) 374–380) states that any exchangeable partition of can be represented as a paintbox based on a random mass-partition. Similarly, any exchangeable composition (i.e., ordered partition of ) can be represented as a paintbox based on an interval-partition (Gnedin (1997) Ann. Probab. 25 (1997) 1437–1450).

Our first main result is that any exchangeable coalescent process (not necessarily Markovian) can be represented as a paintbox based on a random nondecreasing process valued in interval-partitions, called nested interval-partition, generalizing the notion of comb metric space introduced in Lambert and Uribe Bravo (2017) (p-Adic Numbers Ultrametric Anal. Appl. 9 (2017) 22–38) to represent compact ultrametric spaces.

As a special case, we show that any Λ-coalescent can be obtained from a paintbox based on a unique random nested interval partition called Λ-comb, which is Markovian with explicit transitions. This nested interval-partition directly relates to the flow of bridges of Bertoin and Le Gall (2003) (Probab. Theory Related Fields 126 (2003) 261–288). We also display a particularly simple description of the so-called evolving coalescent (Pfaffelhuber and Wakolbinger (2006) Stochastic Process. Appl. 116 (2006) 1836–1859) by a comb-valued Markov process.

Next, we prove that any ultrametric measure space U, under mild measure-theoretic assumptions on U, is the leaf set of a tree composed of a separable subtree called the backbone, on which are grafted additional subtrees, which act as star-trees from the standpoint of sampling. Displaying this so-called weak isometry requires us to extend the Gromov-weak topology of Greven, Pfaffelhuber and Winter (2009) (Probab. Theory Related Fields 145 (2009) 285–322), that was initially designed for separable metric spaces, to nonseparable ultrametric spaces. It allows us to show that for any such ultrametric space U, there is a nested interval-partition which is (1) indistinguishable from U in the Gromov-weak topology; (2) weakly isometric to U if U has a complete backbone; (3) isometric to U if U is complete and separable.


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Félix Foutel-Rodier. Amaury Lambert. Emmanuel Schertzer. "Exchangeable coalescents, ultrametric spaces, nested interval-partitions: A unifying approach." Ann. Appl. Probab. 31 (5) 2046 - 2090, October 2021.


Received: 1 October 2018; Revised: 1 October 2020; Published: October 2021
First available in Project Euclid: 29 October 2021

Digital Object Identifier: 10.1214/20-AAP1641

Primary: 60G09
Secondary: 54E70 , 60C05 , 60J35

Keywords: Combs , compositions , flow of bridges , Gromov-weak topology , metric measure spaces , nested compositions , Λ-coalescents

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.31 • No. 5 • October 2021
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