The Annals of Probability

The representation of composition structures

Alexander V. Gnedin

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A composition structure is a sequence of consistent probability distributions for compositions (ordered partitions) of $n = 1, 2, \dots$. Any composition structure can be associated with an exchangeable random composition of the set of natural numbers. Following Donnelly and Joyce, we study the problem of characterizing a generic composition structure as a convex mixture of the "extreme" ones. We topologize the family $\mathscr{U}$ of open subsets of [0, 1] so that $\mathscr{U}$ becomes compact and show that $\mathscr{U}$ is homeomorphic to the set of extreme composition structures. The general composition struc-ture is related to a random element of $\mathscr{U}$ via a construction introduced by J. Pitman.

Article information

Ann. Probab. Volume 25, Number 3 (1997), 1437-1450.

First available in Project Euclid: 18 June 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G09: Exchangeability
Secondary: 60C05: Combinatorial probability 60J50: Boundary theory

Composition structure partition structure exchangeability paintbox process random set


Gnedin, Alexander V. The representation of composition structures. Ann. Probab. 25 (1997), no. 3, 1437--1450. doi:10.1214/aop/1024404519.

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