Generalized Poisson--Dirichlet Distributions Based on the Dickman Subordinator

Abstract

We study exchangeable random partitions based on an underlying Dickman subordinator and the corresponding family of Poisson--Dirichlet distributions. The large sample distribution of the vector representing the block sizes and the number of blocks in a partition of $\{1,2,\dots,n\}$ is shown to be, after norming and centering, a product of independent Poissons and a normal distribution. In a species or gene sampling situation, these quantities represent the abundances and the numbers of species or genes observed in a sample of size $n$ from the corresponding Poisson--Dirichlet distribution. We include a summary of known convergence results concerning the Dickman subordinator in this context.