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Poisson Representation of a Ewens Fragmentation Process

Published online by Cambridge University Press:  01 November 2007

ALEXANDER GNEDIN
Affiliation:
Mathematical Institute, Utrecht University, The Netherlands (e-mail: [email protected]
JIM PITMAN
Affiliation:
Department of Statistics, University of California, Berkeley, USA (e-mail: [email protected])

Abstract

A simple explicit construction is provided of a partition-valued fragmentation process whose distribution on partitions of [n] = 1,. . .,n at time θ ≥ 0 is governed by the Ewens sampling formula with parameter θ. These partition-valued processes are exchangeable and consistent, as n varies. They can be derived by uniform sampling from a corresponding mass fragmentation process defined by cutting a unit interval at the points of a Poisson process with intensity θx−1dx on/mathbbR+, arranged to beintensifying as θ increases.

Type
Paper
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Arratia, R., Barbour, A. D. and Tavaré, S. (2003) Logarithmic Combinatorial Structures: A Probabilistic Approach, Vol. 1 of EMS Monographs in Mathematics, European Mathematical Society Publishing House, Zürich.CrossRefGoogle Scholar
[2]Berestycki, N. and Pitman, J. (2006) Gibbs distributions for random partitions generated by a fragmentation process. J. Stat. Phys.Google Scholar
[3]Bertoin, J. (2006) Random Fragmentation and Coagulation Processes, Cambridge University Press.CrossRefGoogle Scholar
[4]Chase, K. C. and Mekjian, A. Z. (1994) Nuclear fragmentation and its parallels. Phys. Rev. C 49 21642176.CrossRefGoogle ScholarPubMed
[5]Evans, S. N. and Pitman, J. (1998) Construction of Markovian coalescents. Ann. Inst. Henri Poincaré 34 339383.CrossRefGoogle Scholar
[6]Gnedin, A. (2004) Three sampling formulas. Combin. Probab. Comput. 13 185193.Google Scholar
[7]Gnedin, A. and Pitman, J. (2004) Regenerative partition structures. Electron. J. Combin. 11 12.CrossRefGoogle Scholar
[8]Gnedin, A. and Pitman, J. (2005) Regenerative composition structures. Ann. Probab. 33 445479.Google Scholar
[9]Gnedin, A. and Pitman, J. (2005) Self-similar and Markov composition structures. Representation Theory, Dynamical Systems, Combinatorial and Algorithmic Methods, Part 13 Lodkin, A. A., ed.), Vol. 326 of Zapiski Nauchnyh Seminarov POMI, pp. 5984.Google Scholar
[10]Ignatov, T. (1982) A constant arising in the asymptotic theory of symmetric groups, and Poisson–Dirichlet measures. Theor. Probab. Appl. 27 136147.Google Scholar
[11]Lee, S. J. and Mekjian, A. Z. (1992) Canonical studies of the cluster distribution, dynamical evolution, and critical temperature in nuclear multifragmentation processes. Phys. Rev. C 45 12841310.CrossRefGoogle ScholarPubMed
[12]Mekjian, A. Z. (1991) Cluster distributions in physics and genetic diversity. Phys. Rev. A 44 83618374.CrossRefGoogle ScholarPubMed
[13]Mekjian, A. Z. and Lee, S. J. (1991) Models of fragmentation and partitioning phenomena based on the symmetric group Sn and combinatorial analysis. Phys. Rev. A 44 62946312.CrossRefGoogle Scholar
[14]Pitman, J. (2006) Combinatorial Stochastic Processes, Vol. 1875 of Lecture Notes in Mathematics, Springer.Google Scholar