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...

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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
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cited_by cdi_FETCH-LOGICAL-c385t-21d703f4b3905620db09c6637e51f5f830d07791d9dbbc88c8682c6573247d583
cites cdi_FETCH-LOGICAL-c385t-21d703f4b3905620db09c6637e51f5f830d07791d9dbbc88c8682c6573247d583
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container_title The Annals of probability
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creator Diaconis, Persi
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description 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.
doi_str_mv 10.1214/aop/1079021468
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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
title The Poisson-Dirichlet Law Is the Unique Invariant Distribution for Uniform Split-Merge Transformations
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