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2017 An ergodic theorem for partially exchangeable random partitions
Jim Pitman, Yuri Yakubovich
Electron. Commun. Probab. 22: 1-10 (2017). DOI: 10.1214/17-ECP95

Abstract

We consider shifts $\Pi _{n,m}$ of a partially exchangeable random partition $\Pi _\infty $ of $\mathbb{N} $ obtained by restricting $\Pi _\infty $ to $\{n+1,n+2,\dots , n+m\}$ and then subtracting $n$ from each element to get a partition of $[m]:= \{1, \ldots , m \}$. We show that for each fixed $m$ the distribution of $\Pi _{n,m}$ converges to the distribution of the restriction to $[m]$ of the exchangeable random partition of $\mathbb{N} $ with the same ranked frequencies as $\Pi _\infty $. As a consequence, the partially exchangeable random partition $\Pi _\infty $ is exchangeable if and only if $\Pi _\infty $ is stationary in the sense that for each fixed $m$ the distribution of $\Pi _{n,m}$ on partitions of $[m]$ is the same for all $n$. We also describe the evolution of the frequencies of a partially exchangeable random partition under the shift transformation. For an exchangeable random partition with proper frequencies, the time reversal of this evolution is the heaps process studied by Donnelly and others.

Citation

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Jim Pitman. Yuri Yakubovich. "An ergodic theorem for partially exchangeable random partitions." Electron. Commun. Probab. 22 1 - 10, 2017. https://doi.org/10.1214/17-ECP95

Information

Received: 2 July 2017; Accepted: 24 October 2017; Published: 2017
First available in Project Euclid: 23 November 2017

zbMATH: 06827046
MathSciNet: MR3734103
Digital Object Identifier: 10.1214/17-ECP95

Subjects:
Primary: 37A30, 60B99, 60G09, 60J10

JOURNAL ARTICLE
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