# Exchangeable and partially exchangeable random partitions

@article{Pitman1995ExchangeableAP, title={Exchangeable and partially exchangeable random partitions}, author={Jim Pitman}, journal={Probability Theory and Related Fields}, year={1995}, volume={102}, pages={145-158} }

SummaryCall a random partition of the positive integerspartially exchangeable if for each finite sequence of positive integersn1,...,nk, the probability that the partition breaks the firstn1+...+nk integers intok particular classes, of sizesn1,...,nk in order of their first elements, has the same valuep(n1,...,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(n1,...nk). A…

## 526 Citations

Characterizations of exchangeable partitions and random discrete distributions by deletion properties

- Mathematics
- 2009

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…

The distribution of the number of distinct values in a finite exchangeable sequence

- Mathematics
- 2021

Let Kn denote the number of distinct values among the first n terms of an infinite exchangeable sequence of random variables (X1, X2, . . .). We prove for n = 3 that the extreme points of the convex…

Extreme sizes in Gibbs-type exchangeable random partitions

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- 2013

Gibbs-type exchangeable random partition, which is a class of multiplicative measures on the set of positive integer partitions, appear in various contexts, including Bayesian statistics, random…

Coherent random allocations, and the Ewens-Pitman formula

- Mathematics
- 2006

Assume that there is a random number K of positive integer random variables S1, …, SK that are conditionally independent given K and all have identical distributions. A random integer partition N =…

Binary sequential representations of random partitions

- Mathematics, Computer Science
- 2005

A two-parameter generalization of Ewens' partition is obtained by considering random partitions constructed from discrete renewal processes and introducing a convolution-type product on 0-1 sequences.

Increments of Random Partitions

- Mathematics
- 2008

For any partition of {1, 2,. .. , n} we define its increments Xi, 1 ≤ i ≤ n by Xi = 1 if i is the smallest element in the partition block that contains it, Xi = 0 otherwise. We prove that for…

Increments of Random Partitions

- Mathematics, Computer ScienceCombinatorics, Probability and Computing
- 2006

The Chinese Restaurant Process CRP (the partition with distribution given by the Ewens sampling formula with parameter $\theta$) is the only exchangeable random partition with independent increments because of the law of the increments.

Frequency of Frequencies Distributions and Size-Dependent Exchangeable Random Partitions

- Mathematics
- 2016

ABSTRACT Motivated by the fundamental problem of modeling the frequency of frequencies (FoF) distribution, this article introduces the concept of a cluster structure to define a probability function…

Ordered sizes in exchangeable random partitions and their asymptotics

- Mathematics
- 2013

Some distributional results on ordered sizes in exchangeable random partitions of a natural number are obtained by combinatorial arguments. Analysis of generating functions yields their asymptotics.…

Record Indices and Age-Ordered Frequencies in Exchangeable Gibbs Partitions

- Mathematics
- 2007

The frequencies of an exchangeable Gibbs random partition of the integers (Gnedin and Pitman 2005) are considered in their age-order, i.e. their size-biased order. We study their dependence on the…

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