A dynamical characterization of Poisson-Dirichlet distributions

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 $PD(\alpha,0), \alpha\in(0,1)$.