Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T13:49:16.678Z Has data issue: false hasContentIssue false
  • English
  • Français

Poisson Representation of a Ewens Fragmentation Process

Published online by Cambridge University Press:  01 November 2007

ALEXANDER GNEDIN  and
JIM PITMAN
ALEXANDER GNEDIN
Affiliation:
Mathematical Institute, Utrecht University, The Netherlands (e-mail: gnedin@math.uu.nl
JIM PITMAN
Affiliation:
Department of Statistics, University of California, Berkeley, USA (e-mail: pitman@stat.Berkeley.EDU)
Get access Rights & Permissions [Opens in a new window]

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
Information
Combinatorics, Probability and Computing , Volume 16 , Issue 6 , November 2007 , pp. 819 - 827
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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 2164–2176.CrossRefGoogle ScholarPubMed
[5]Evans, S. N. and Pitman, J. (1998) Construction of Markovian coalescents. Ann. Inst. Henri Poincaré 34 339–383.CrossRefGoogle Scholar
[6]Gnedin, A. (2004) Three sampling formulas. Combin. Probab. Comput. 13 185–193.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. 33 445–479.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. 59–84.Google Scholar
[10]Ignatov, T. (1982) A constant arising in the asymptotic theory of symmetric groups, and Poisson–Dirichlet measures. Theor. Probab. Appl. 27 136–147.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 1284–1310.CrossRefGoogle ScholarPubMed
[12]Mekjian, A. Z. (1991) Cluster distributions in physics and genetic diversity. Phys. Rev. A 44 8361–8374.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 6294–6312.CrossRefGoogle Scholar
[14]Pitman, J. (2006) Combinatorial Stochastic Processes, Vol. 1875 of Lecture Notes in Mathematics, Springer.Google Scholar