The Poisson-Dirichlet law is the unique invariant distribution for uniform split-merge transformations
Probability
2016-09-07 v2
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.
Cite
@article{arxiv.math/0305313,
title = {The Poisson-Dirichlet law is the unique invariant distribution for uniform split-merge transformations},
author = {Persi Diaconis and Eddy Mayer-Wolf and Ofer Zeitouni and Martin Zerner},
journal= {arXiv preprint arXiv:math/0305313},
year = {2016}
}
Comments
To appear in Annals Probab. 6 figures Only change in new version is addition of proof (at end of article) that the state (1,0,0,...) is transient