A dynamical characterization of Poisson-Dirichlet distributions
Louis-Pierre Arguin, Princeton University
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 ξ_1>= ξ_2 >= ...>= 0 and ∑i ξi=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) → ({ξi Wi}/{∑j ξj Wj},i∈ N)
where the weights are reordered after evolution,
then it must be a mixture of Poisson-Dirichlet distributions PD(α,0), α∈(0,1).
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