Characterizations of exchangeable partitions and random discrete distributions by deletion properties

We prove a long-standing conjecture which characterises the Ewens-Pitman two- parameter 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 indi- viduals is chosen uniformly at random, independently of the random partitionn of these individuals into various types, and all individuals of the same type as the cho- sen individual are deleted, then for each r > 0, given that r individuals remain, these individuals are partitioned according to � 0 for some sequence of random partitions (� 0 ) which does not depend on n. An analogous result characterizes the associ- ated Poisson-Dirichlet family of random discrete distributions by an independence property related to random deletion of a frequency chosen by a size-biased pick. We also survey the regenerative properties of members of the two-parameter family, and settle a question regarding the explicit arrangement of intervals with lengths given by the terms of the Poisson-Dirichlet random sequence into the interval partition induced by the range of a homogeneous neutral-to-the right process.

• Publication year

  2009

• Venue

  Unknown venue

• Publication date

  2009-09-20

• Fields of study

  Mathematics

• Identifiers
  DOI 10.1017/CBO9781139107174.013arXiv 0909.3642
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

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