A dynamical characterization of Poisson-Dirichlet distributions

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(alpha, 0). Precisely, let xi be a proper random mass-partition i.e. a random sequence (xi(i), i is an element of N) such that xi(1) >= xi(2) >= ... and Sigma(i) xi(i) = 1 a.s. Consider {W-i}(i is an element of N), an iid sequence of random positive numbers whose distribution is absolutely continuous with respect to the Lebesgue measure and E[W-lambda] < infinity for all lambda is an element of R. It is shown that, if the law of xi is invariant under the random reshuffling (xi(i), i is an element of N) -> (xi W-i(i)/Sigma(j)xi W-j(j), i is an element of N) where the weights are reordered after evolution, then it must be a mixture of Poisson-Dirichlet distribution PD(alpha, 0), alpha is an element of (0, 1).