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(α,0). Precisely, let ξ be a proper random mass-partition i.e. a random sequence (ξi,i∈N) such that ξ1≥ξ2≥⋯≥0 and ∑iξi=1 a.s. Consider {Wi}i∈N, an iid sequence of random positive numbers whose distribution is absolutely continuous with respect to the Lebesgue measure and E[Wλ]<∞ for all λ∈R. It is shown that, if the law of ξ is invariant under the random reshuffling (ξi,i∈N)→(ξiWi∑jξjWj,i∈N) where the weights are reordered after evolution, then it must be a mixture of Poisson-Dirichlet distributions PD(α,0),α∈.
Citation
Louis-Pierre Arguin. "A dynamical characterization of Poisson-Dirichlet distributions." Electron. Commun. Probab. 12 283 - 290, 2007. https://doi.org/10.1214/ECP.v12-1300
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