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A Consistent Markov Partition Process Generated from the Paintbox Process

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Harry Crane
Harry Crane*
Affiliation:
University of Chicago
*
∗ Postal address: Department of Statistics, University of Chicago, Eckhart Hall Room 108, 5734 South University Avenue, Chicago, IL 60637, USA. Email address: crane@uchicago.edu
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Abstract

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We study a family of Markov processes on P (k), the space of partitions of the natural numbers with at most k blocks. The process can be constructed from a Poisson point process on R + x ∏ i=1 k P (k) with intensity dt ⊗ ϱν (k), where ϱν is the distribution of the paintbox based on the probability measure ν on P m, the set of ranked-mass partitions of 1, and ϱν (k) is the product measure on ∏ i=1 k P (k). We show that these processes possess a unique stationary measure, and we discuss a particular set of reversible processes for which transition probabilities can be written down explicitly.

Keywords

Paintbox processEwens' partitionPoisson-Dirichlet distributionpartition process

MSC classification

Secondary: 60J25: Continuous-time Markov processes on general state spaces 60G09: Exchangeability
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 3 , September 2011 , pp. 778 - 791
Copyright
Copyright © Applied Probability Trust 2011 

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