Electronic Communications in Probability

A dynamical characterization of Poisson-Dirichlet distributions

Louis-Pierre Arguin

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Abstract

We show that a slight modification of a theorem of Ruzmaikina and Aizenman on competing particle systems on the real line leads to a characterization of Poisson-Dirichlet distributions $PD(\alpha,0)$. Precisely, let $\xi$ be a proper random mass-partition i.e. a random sequence $(\xi_i,i\in N)$ such that $\xi_1 \geq \xi_2 \geq \dots \geq 0$ and $\sum_i \xi_i =1$ a.s. Consider $\{W_i\}_{i\in N}$, an iid sequence of random positive numbers whose distribution is absolutely continuous with respect to the Lebesgue measure and $E[W^\lambda]<\infty$ for all $\lambda \in R$. It is shown that, if the law of $\xi$ is invariant under the random reshuffling $$( \xi_i , i \in N) \to \left(\frac{\xi_i W_i}{\sum_j \xi_jW_j } , i \in N \right)$$ where the weights are reordered after evolution, then it must be a mixture of Poisson-Dirichlet distributions $PD(\alpha,0), \alpha\in$.

Article information

Source
Electron. Commun. Probab., Volume 12 (2007), paper no. 28, 283-290.

Dates
Accepted: 21 September 2007
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465224971

Digital Object Identifier
doi:10.1214/ECP.v12-1300

Mathematical Reviews number (MathSciNet)
MR2342707

Zentralblatt MATH identifier
1128.60037

Subjects
Primary: 60G55: Point processes
Secondary: 60G57: Random measures

Keywords
Point processes Poisson-Dirichlet distributions

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Arguin, Louis-Pierre. A dynamical characterization of Poisson-Dirichlet distributions. Electron. Commun. Probab. 12 (2007), paper no. 28, 283--290. doi:10.1214/ECP.v12-1300. https://projecteuclid.org/euclid.ecp/1465224971


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References

  • Bertoin, Jean. Random fragmentation and coagulation processes.Cambridge Studies in Advanced Mathematics, 102. Cambridge University Press, Cambridge, 2006. viii+280 pp. ISBN: 978-0-521-86728-3; 0-521-86728-2
  • Dembo, Amir; Zeitouni, Ofer. Large deviations techniques and applications.Second edition.Applications of Mathematics (New York), 38. Springer-Verlag, New York, 1998. xvi+396 pp. ISBN: 0-387-98406-2
  • Diaconis, Persi; Mayer-Wolf, Eddy; Zeitouni, Ofer; Zerner, Martin P. W. The Poisson-Dirichlet law is the unique invariant distribution for uniform split-merge transformations. Ann. Probab. 32 (2004), no. 1B, 915–938.
  • Liggett, Thomas M. Random invariant measures for Markov chains, and independent particle systems. Z. Wahrsch. Verw. Gebiete 45 (1978), no. 4, 297–313.
  • Ruzmaikina, Anastasia; Aizenman, Michael. Characterization of invariant measures at the leading edge for competing particle systems. Ann. Probab. 33 (2005), no. 1, 82–113.