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Exchangeable and partially exchangeable random partitions. (English) Zbl 0821.60047

Summary: Call a random partition of the positive integers partially exchangeable if for each finite sequence of positive integers \(n_ 1, \dots, n_ k\), the probability that the partition breaks the first \(n_ 1 + \cdots + n_ k\) integers into \(k\) particular classes, of sizes \(n_ 1, \dots, n_ k\) in order of their first elements, has the same value \(p(n_ 1, \dots, n_ k)\) for every possible choice of classes subject to the sizes constraint. A random partition is exchangeable iff it is partially exchangeable for a symmetric function \(p(n_ 1, \dots, n_ k)\). A representation is given for partially exchangeable random partitions which provides a useful variation of Kingman’s representation in the exchangeable case. Results are illustrated by the two-parameter generalization of Ewens’ partition structure.

MSC:

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