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Exchangeable coalescents, ultrametric spaces, nested interval-partitions: A unifying approach. (arXiv:1807.05165v1 [math.PR])
来源于:arXiv
Kingman (1978)'s representation theorem 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 查看全文>>