Partially Exchangeable Random Partitions
نویسنده
چکیده
A random partition of the positive integers is called partially exchangeable if for each finite sequence of positive integers n,, ... , nk, the probability that the partition breaks the first n1 + + nk integers into k particular classes, of sizes nl,.. . ., nk in order of their first elements, has the same value p(nl,...,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 function p(nl, . . ., nk). A representation is given for partially exchangeable random partitions that is similar to Kingman's representation in the exchangeable case. These representations are viewed as variations of de Finetti's representation of exchangeable sequences, and as identifications of the Martin boundary of associated Markov chains. In the exchangeable case, information is provided about the joint distribution of the proportions of classes in order of their appearance. This gives a constraint on the finite dimensional distributions of a random discrete probability distribution on the positive integers that is equivalent to invariance under size-biased random permutation. The results are illustrated by the two-parameter generalization of Ewens' partition structure. *Research supported by N.S.F. Grant MCS91-07531
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Exchangeable and partially exchangeable random partitions
Call a random partition of the positive integers partially exchangeable if for each finite sequence of positive integers n l , . . . , nk, the probability that the partition breaks the first nl + . . . § nk integers into k particular classes, of sizes nl . . . . , nk in order of their first elements, has the same value p(na . . . . . nk) for every possible choice of classes subject to the sizes...
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