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January 2004 The Poisson-Dirichlet law is the unique invariant distribution for uniform split-merge transformations
Persi Diaconis, Eddy Mayer-Wolf, Ofer Zeitouni, Martin P. W. Zerner
Ann. Probab. 32(1B): 915-938 (January 2004). DOI: 10.1214/aop/1079021468

Abstract

We consider a Markov chain on the space of (countable) partitions of the interval $[0,1]$, obtained first by size-biased sampling twice (allowing repetitions) and then merging the parts (if the sampled parts are distinct) or splitting the part uniformly (if the same part was sampled twice). We prove a conjecture of Vershik stating that the Poisson--Dirichlet law with parameter $\theta=1$ is the unique invariant distribution for this Markov chain. Our proof uses a combination of probabilistic, combinatoric and representation-theoretic arguments.

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Persi Diaconis. Eddy Mayer-Wolf. Ofer Zeitouni. Martin P. W. Zerner. "The Poisson-Dirichlet law is the unique invariant distribution for uniform split-merge transformations." Ann. Probab. 32 (1B) 915 - 938, January 2004. https://doi.org/10.1214/aop/1079021468

Information

Published: January 2004
First available in Project Euclid: 11 March 2004

zbMATH: 1049.60088
MathSciNet: MR2044670
Digital Object Identifier: 10.1214/aop/1079021468

Subjects:
Primary: 60K35
Secondary: 60G55, 60J27

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 1B • January 2004
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