EXCHANGEABLE COALESCENTS, ULTRAMETRIC SPACES, NESTED INTERVAL-PARTITIONS: A UNIFYING APPROACH

Kingman's (1978) representation theorem (J. Lond. Math. Soc. (2) 18 (1978) 374-380) 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) 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 intervalpartition, 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 Lambda-coalescent can be obtained from a paintbox based on a unique random nested interval partition called Lambda-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.