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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

Citations:

Zbl 1096.60042
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