SMILE

Stochastic Models for the Inference of Life Evolution

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

Foutel-Rodier, F., Lambert, A., Schertzer, E.

arXiv:1807.05165

2018

Kingman's representation theorem (Kingman 1978) states that any exchangeable partition of \$$\mathbb{N}\$$ can be represented as a paintbox based on a random mass-partition. Similarly, any exchangeable composition (i.e.\ ordered partition of \$$\mathbb{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 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, 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.

Bibtex

@article{foutel2018exchangeable,
title={Exchangeable coalescents, ultrametric spaces, nested interval-partitions: A unifying approach},
author={Foutel-Rodier, F{\'e}lix and Lambert, Amaury and Schertzer, Emmanuel},
journal={arXiv:1807.05165},
eprint={arXiv:1807.05165},
url={https://arxiv.org/pdf/1807.05165},
year={2018}
}

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