Generalized Poisson–Dirichlet distributions based on the Dickman subordinator

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.