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Characterizations of exchangeable partitions and random discrete distributions by deletion properties
, 2009
"... We prove a longstanding conjecture which characterises the EwensPitman twoparameter family of exchangeable random partitions, plus a short list of limit and exceptional cases, by the following property: for each n = 2,3,..., if one of n individuals is chosen uniformly at random, independently of ..."
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We prove a longstanding conjecture which characterises the EwensPitman twoparameter family of exchangeable random partitions, plus a short list of limit and exceptional cases, by the following property: for each n = 2,3,..., if one of n individuals is chosen uniformly at random, independently of the random partition πn of these individuals into various types, and all individuals of the same type as the chosen individual are deleted, then for each r> 0, given that r individuals remain, these individuals are partitioned according to π ′ r for some sequence of random partitions (π ′ r) that does not depend on n or r. An analogous result characterizes the associated PoissonDirichlet family of random discrete distributions by an independence property related to random deletion of a frequency chosen by a sizebiased pick. We also survey the regenerative properties of members of the twoparameter family, and settle a question regarding the explicit arrangement of intervals with lengths given by the terms of the PoissonDirichlet random sequence into the interval partition induced by the range of a neutraltothe right process.
Some Diffusion Processes Associated With Two Parameter PoissonDirichlet Distribution and Dirichlet Process
, 903
"... The two parameter PoissonDirichlet distribution PD(α,θ) is the distribution of an infinite dimensional random discrete probability. It is a generalization of Kingman’s PoissonDirichlet distribution. The two parameter Dirichlet process Πα,θ,ν0 is the law of a pure atomic random measure with masses ..."
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The two parameter PoissonDirichlet distribution PD(α,θ) is the distribution of an infinite dimensional random discrete probability. It is a generalization of Kingman’s PoissonDirichlet distribution. The two parameter Dirichlet process Πα,θ,ν0 is the law of a pure atomic random measure with masses following the two parameter PoissonDirichlet distribution. In this article we focus on the construction and the properties of the infinite dimensional symmetric diffusion processes with respective symmetric measures PD(α,θ) and Πα,θ,ν0. The methods used come from the theory of Dirichlet forms. 1