Generalized Poisson--Dirichlet Distributions Based on the Dickman Subordinator

Abstract

We derive large-sample and other limiting distributions of the "frequency of frequencies" vector, Mn, together with the number of species, Kn, in a Poisson-Dirichlet or generalised Poisson-Dirichlet gene or species sampling model. Models analysed include those constructed from gamma and α-stable subordinators by Kingman, the two parameter extension by Pitman and Yor, and another two parameter version constructed by omitting large jumps from an α-stable subordinator. In the Poisson-Dirichlet case Mn and Kn turn out to be asymptotically independent, and notable, especially for statistical applications, is that in other cases the conditional limiting distribution of Mn, given Kn, is normal, after certain centering and norming.