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.