On the Poisson–Dirichlet Limit

Kingman showed that if the vectorXNis distributed according to the Dirichlet law then the vector of descending order statistics converges, under certain conditions, to a nondegenerate limit. This contrasts with the fact that the limit of any fixed component ofXNis zero. Nevertheless,XNdoes have, in some sense, a nondegenerate limit which we identify with a random interval partition. Convergence of this kind does not require rearranging of the components and implies the existence of limit distributions for a class of functionals which are not covered by the Kingman result.