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A representation of exchangeable hierarchies by sampling from random real trees

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A hierarchy on a set S, also called a total partition of S, is a collection $$\mathcal {H}$$H of subsets of S such that $$S \in \mathcal {H}$$S∈H, each singleton… Click to show full abstract

A hierarchy on a set S, also called a total partition of S, is a collection $$\mathcal {H}$$H of subsets of S such that $$S \in \mathcal {H}$$S∈H, each singleton subset of S belongs to $$\mathcal {H}$$H, and if $$A, B \in \mathcal {H}$$A,B∈H then $$A \cap B$$A∩B equals either A or B or $$\varnothing $$∅. Every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy $$\mathcal {H}$$H associated as follows with a random real tree $$\mathcal {T}$$T equipped with root element 0 and a random probability distribution p on the Borel subsets of $$\mathcal {T}$$T: given $$(\mathcal {T},p)$$(T,p), let $$t_1,t_2, \ldots $$t1,t2,… be independent and identically distributed according to p, and let $$\mathcal {H}$$H comprise all singleton subsets of $${\mathbb {N}}$$N, and every subset of the form $$\{j:t_j \in F(x)\}$${j:tj∈F(x)} as x ranges over $$\mathcal {T}$$T, where F(x) is the fringe subtree of $$\mathcal {T}$$T rooted at x. There is also the alternative characterization: every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy $$\mathcal {H}$$H derived as follows from a random hierarchy $${\mathscr {H}}$$H on [0, 1] and a family $$(U_j)$$(Uj) of i.i.d. Uniform [0,1] random variables independent of $${\mathscr {H}}$$H: let $$\mathcal {H}$$H comprise all sets of the form $$\{j:U_j \in B\}$${j:Uj∈B} as B ranges over the members of $${\mathscr {H}}$$H.

Keywords: random hierarchy; representation exchangeable; exchangeable hierarchies; random; random real

Journal Title: Probability Theory and Related Fields
Year Published: 2018

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