The Poisson-Dirichlet law is the unique invariant distribution for uniform split-merge transformations

• Persi Diaconis ^[2] ; Eddy Mayer-Wolf ^[1] ; Ofer Zeitouni ^[3] ; Martin P. W. Zerner ^[2]
  1. [1] Technion – Israel Institute of Technology

     Technion – Israel Institute of Technology

     Israel

     
  2. [2] Standford University
  3. [3] University of Minesota and Technion

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 32, Nº. 1, 2, 2004, págs. 915-938
• Idioma: inglés
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• Resumen

  • 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 θ=1 is the unique invariant distribution for this Markov chain. Our proof uses a combination of probabilistic, combinatoric and representation-theoretic arguments.