A dynamical characterization of Poisson-Dirichlet distributions

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 $PD(a,0)$. Precisely, let $s$ be a proper random mass-partition i.e. a random sequence $(s_i,i\in \mathbb{N})$ such that $s_1\ge s_2\ge ...$ and ${\sum }_i{s}_i=1$ a.s. Consider ${W_i}_{i\in \mathbb{N}}$, an iid sequence of positive random variables with a density and such that $E[W^{\lambda }]$ is finite for all $\lambda \in \mathbb{R}$. 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 $PD(a,0)$, $a\in (0,1)$.

8 pages; final version accepted for publication