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A Consistent Markov Partition Process Generated from the Paintbox Process

Part of: Markov processes Stochastic 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: [email protected]
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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 

References

[1] Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5, 3–48.Google Scholar
[2] Berestycki, J. (2004). Exchangeable fragmentation-coalescence processes and their equilibrium measures. Electron. J. Prob. 9, 770–824.Google Scholar
[3] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge University Press.Google Scholar
[4] Bertoin, J. (2008). Two-parameter Poisson-Dirichlet measures and reversible exchangeable fragmentation-coagulation processes. Combinatorics Prob. Comput. 17, 329–337.Google Scholar
[5] Bertoin, J. (2010). Exchangeable Coalescents (Lecture Notes for PIMS Summer School in Probability). University of Washington and Microsoft Research.Google Scholar
[6] Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York.Google Scholar
[7] Blei, D., Ng, A. and Jordan, M. (2003). Latent Dirichlet allocation. J. Machine Learning Res. 3, 993–1022.Google Scholar
[8] Burke, C. J. and Rosenblatt, M. (1958). A Markovian function of a Markov chain. Ann. Math. Statist. 29, 1112–1122.Google Scholar
[9] Efron, B. and Thisted, R. (1976). Estimating the number of unseen species: how many words did Shakespeare know? Biometrika 63, 435–447.Google Scholar
[10] Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theoret. Pop. Biol. 3, 87–112.Google Scholar
[11] Fisher, R. A., Corbet, A. S. and Williams, C. B. (1943). The relation between the number of species and the number of individuals in a random sample of an animal population. J. Animal Ecology 12, 42–58.Google Scholar
[12] Kingman, J. F. C. (1978). Random partitions in population genetics. Proc. R. Soc. London A 361, 1–20.Google Scholar
[13] McCullagh, P. (2010). Random permutations and partition models. In International Encyclopedia of Statistical Science, ed. Lovric, M., Springer, Berlin, pp. 1170–1177.Google Scholar
[14] McCullagh, P. and Møller, J. (2005). The permanent process. Adv. Appl. Prob. 38, 873–888.Google Scholar
[15] McCullagh, P. and Yang, J. (2008). How many clusters? Bayesian Anal. 3, 101–120.Google Scholar
[16] McCullagh, P., Pitman, J. and Winkel, M. (2008). Gibbs fragmentation trees. Bernoulli 14, 988–1002.Google Scholar
[17] Pitman, J. (2006). Combinatorial Stochastic Processes (Lecture Notes Math. 1875). Springer, New York.Google Scholar