Asymptotics of certain coagulation-fragmentation processes and invariant Poisson-Dirichlet measures

We consider Markov chains 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 with probability \(\beta_m\) (if the sampled parts are distinct) or splitting the part with probability \(\beta_s...

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Bibliographic Details
Published in:arXiv.org 2001-05
Main Authors: Mayer-Wolf, Eddy, Zeitouni, Ofer, Zerner, Martin P W
Format: Article
Language:English
Subjects:
Coagulation
Dirichlet problem
Invariants
Markov chains
Online Access:Get full text
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Summary:We consider Markov chains 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 with probability \(\beta_m\) (if the sampled parts are distinct) or splitting the part with probability \(\beta_s\) according to a law \(\sigma\) (if the same part was sampled twice). We characterize invariant probability measures for such chains. In particular, if \(\sigma\) is the uniform measure then the Poisson-Dirichlet law is an invariant probability measure, and it is unique within a suitably defined class of ``analytic'' invariant measures. We also derive transience and recurrence criteria for these chains.
ISSN:2331-8422