Exchangeable coalescents, ultrametric spaces, nested interval-partitions: A unifying approach

Abstract : Kingman (1978)'s representation theorem states that any exchangeable partition of N can be represented as a paintbox based on a random mass-partition. Similarly, any exchangeable composition (i.e. ordered partition of N can be represented as a paintbox based on an interval-partition (Gnedin 1997). Our first main result is that any exchangeable coalescent process (not necessarily Markovian) can be represented as a paintbox based on a random non-decreasing process valued in interval-partitions, called nested interval-partition, generalizing the notion of comb metric space introduced by Lambert & Uribe Bravo (2017) to represent compact ultrametric spaces. As a special case, we show that any Lambda-coalescent can be obtained from a paintbox based on a unique random nested interval partition called Lambda-comb, which is Markovian with explicit semi-group. This nested interval-partition directly relates to the flow of bridges of Bertoin & Le Gall (2003). We also display a particularly simple description of the so-called evolving coalescent (Pfaffelhuber & Wakolbinger 2006) by a comb-valued Markov process. Next, we prove that any measured ultrametric 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 et al (2006), that was initially designed for separable metric spaces, to non-separable 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 complete backbone; 3) isometric to U if U is complete and separable.
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Pré-publication, Document de travail
38 pages, 7 figures. 2018
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Dernière modification le : mardi 19 mars 2019 - 01:23:29
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  • HAL Id : hal-01841003, version 1
  • ARXIV : 1807.05165

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Félix Foutel-Rodier, Amaury Lambert, Emmanuel Schertzer. Exchangeable coalescents, ultrametric spaces, nested interval-partitions: A unifying approach. 38 pages, 7 figures. 2018. 〈hal-01841003〉

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