, Volume 102, Issue 2, pp 145-158

Exchangeable and partially exchangeable random partitions

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Call a random partition of the positive integerspartially exchangeable if for each finite sequence of positive integersn 1,...,n k, the probability that the partition breaks the firstn 1+...+nk integers intok particular classes, of sizesn 1,...,nk in order of their first elements, has the same valuep(n 1,...,nk) for every possible choice of classes subject to the sizes constraint. A random partition is exchangeable iff it is partially exchangeable for a symmetric functionp(n 1,...nk). 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.

Research supported by N.S.F. Grants MCS91-07531 and DMS-9404345