A dynamical characterization of Poisson-Dirichlet distributions.(English)Zbl 1128.60037

Summary: We show that a slight modification of a theorem of A. Ruzmaikina and M. Aizenman [Ann. Probab. 33, No. 1, 82–113 (2005; Zbl 1096.60042)] on competing particle systems on the real line leads to a characterization of Poisson-Dirichlet distributions $$\text{PD}(\alpha,0)$$. Precisely, let $$\xi$$ be a proper random mass-partition, i.e., a random sequence $$(\xi_i$$, $$i\in\mathbb N)$$ such that $$\xi_1\geq\xi_2\geq\dots$$ and $$\sum_i\xi_i=1$$ a.s. Consider $$\{W_i\}_{i\in\mathbb N}$$, an i.i.d. sequence of random positive numbers whose distribution is absolutely continuous with respect to the Lebesgue measure and $$\mathbb E[W^\lambda]<\infty$$ for all $$\lambda\in\mathbb R$$. It is shown that, if the law of $$\xi$$ is invariant under the random reshuffling $(\xi_i,i\in\mathbb N)\mapsto \biggl( \frac{\xi_iW_i}{\sum_j\xi_jW_j},\;i\in\mathbb N\biggr),$ where the weights are reordered after evolution, then it must be a mixture of Poisson-Dirichlet distributions $$\text{PD}(\alpha,0)$$, $$\alpha\in(0,1)$$.

MSC:

 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60G57 Random measures 60K35 Interacting random processes; statistical mechanics type models; percolation theory

Zbl 1096.60042
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