A dynamical characterization of Poisson-Dirichlet distributions

Louis-Pierre Arguin

Published 2007-03-25, updated 2007-09-21Version 3

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 $(s_i, i\in\N)$ such that $s_1\geq s_2\geq ...$ and $\sum_i s_i=1$ a.s. Consider ${W_i}_{i\in\N}$, an iid sequence of positive random variables with a density and such that $E[W^\lambda]$ is finite for all $\lambda\in\R$. It is shown that if the law of $s$ is invariant under a random multiplicative shift $s_i W_i$ of the atoms followed by a renormalization, then it must be a mixture of Poisson-Dirichlet distribution $PD(a,0)$, $a\in (0,1)$.