A dynamical characterization of Poisson-Dirichlet distributions

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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(a,0). Precisely, let s be a proper random mass-partition i.e. a random sequence (sᵢ, i∈) such that s₁≥ s₂≥... and ∑ᵢ sᵢ=1 a.s. Consider Wᵢᵢ∈, an iid sequence of positive random variables with a density and such that E[Wλ] is finite for all λ∈. It is shown that if the law of s is invariant under a random multiplicative shift sᵢ Wᵢ of the atoms followed by a renormalization, then it must be a mixture of Poisson-Dirichlet distribution PD(a,0), a∈ (0,1).