The Poisson-Dirichlet Law Is the Unique Invariant Distribution for Uniform Split-Merge Transformations

We consider a Markow 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...

Full description

Saved in:
Bibliographic Details
Published in:The Annals of probability 2004-01, Vol.32 (1), p.915-938
Main Authors: Diaconis, Persi, Mayer-Wolf, Eddy, Zeitouni, Ofer, Martin P. W. Zerner
Format: Article
Language:English
Subjects:
60G55
60J27
60K35
Atoms
coagulation
Distribution theory
Exact sciences and technology
fragmentation
General topics
Integers
invariant measures
Markov chains
Markov processes
Mathematics
Partitions
Poisson > Dirichlet
Probabilities
Probability and statistics
Probability theory and stochastic processes
Property partitioning
Riemann surfaces
Sampling distributions
Sciences and techniques of general use
Statistics
Uniform laws
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We consider a Markow 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.
ISSN:0091-1798
2168-894X
DOI:10.1214/aop/1079021468