A representation of exchangeable hierarchies by sampling from random real trees

• Published in: Probability theory and related fields 2018-10, Vol.172 (1-2), p.1-29
• Main Authors: Forman, Noah, Haulk, Chris, Pitman, Jim
• Format: Article
• Language: English
• Subjects: Economics; Finance; Hierarchies; Independent variables; Insurance; Integers; Management; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Probability; Probability Theory and Stochastic Processes; Quantitative Finance; Random variables; Statistics for Business; Theoretical
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-017-0799-4

A hierarchy on a set S , also called a total partition of S , is a collection H of subsets of S such that S ∈ H , each singleton subset of S belongs to H , and if A , B ∈ H then A ∩ B equals either A or B or ∅ . Every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy H associated as follows with a random real tree T equipped with root element 0 and a random probability distribution p on the Borel subsets of T : given ( T , p ) , let t 1 , t 2 , … be independent and identically distributed according to p , and let H comprise all singleton subsets of N , and every subset of the form { j : t j ∈ F ( x ) } as x ranges over T , where F ( x ) is the fringe subtree of 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 H derived as follows from a random hierarchy H on [0, 1] and a family ( U j ) of i.i.d. Uniform [0,1] random variables independent of H : let H comprise all sets of the form { j : U j ∈ B } as B ranges over the members of H .