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The representation of composition structures. (English) Zbl 0895.60037

Summary: 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 P. Donnelly and P. Joyce [Adv. Appl. Probab. 23, No. 2, 229-258 (1991; Zbl 0724.60040)], we study the problem of characterizing a generic composition structure as a convex mixture of the “extreme” ones. We topologize the family \({\mathcal U}\) of open subsets of \([0,1]\) so that \({\mathcal U}\) becomes compact and show that \({\mathcal U}\) is homeomorphic to the set of extreme composition structures. The general composition structure is related to a random element of \({\mathcal U}\) via a construction introduced by J. Pitman [Probab. Theory Relat. Fields 102, No. 2, 145-158 (1995; Zbl 0821.60047)].

MSC:

60G09 Exchangeability for stochastic processes
60C05 Combinatorial probability
60J50 Boundary theory for Markov processes
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