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Reversible Markov structures on divisible set partitions

Part of: Probability theory on algebraic and topological structures Markov processes Combinatorial probability

Published online by Cambridge University Press:  30 March 2016

Harry Crane  and
Peter McCullagh
Harry Crane*
Affiliation:
Rutgers University
Peter McCullagh*
Affiliation:
University of Chicago
*
∗ Postal address: Department of Statistics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA. Email address: hcrane@stat.rutgers.edu
∗∗ Postal address: Department of Statistics, University of Chicago, Eckhart Hall, 5734 S. University Avenue, Chicago, IL 60637, USA.
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Abstract

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We study k-divisible partition structures, which are families of random set partitions whose block sizes are divisible by an integer k = 1, 2, …. In this setting, exchangeability corresponds to the usual invariance under relabeling by arbitrary permutations; however, for k > 1, the ordinary deletion maps on partitions no longer preserve divisibility, and so a random deletion procedure is needed to obtain a partition structure. We describe explicit Chinese restaurant-type seating rules for generating families of exchangeable k-divisible partitions that are consistent under random deletion. We further introduce the notion of Markovian partition structures, which are ensembles of exchangeable Markov chains on k-divisible partitions that are consistent under a random process of Markovian deletion. The Markov chains we study are reversible and refine the class of Markov chains introduced in Crane (2011).

Keywords

Markovian partition structureexchangeable partition structureEwens-Pitman partitionChinese restaurant processdivisible partitiongroup-divisible association scheme

MSC classification

Primary: 60C05: Combinatorial probability 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60B99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 3 , September 2015 , pp. 622 - 635
Copyright
Copyright © 2015 by the Applied Probability Trust 

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