Electronic Communications in Probability

A dynamical characterization of Poisson-Dirichlet distributions

Louis-Pierre Arguin

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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Point processes Poisson-Dirichlet distributions

This work is licensed under aCreative Commons Attribution 3.0 License.


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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