Limiting Distributions of Generalised Poisson–Dirichlet Distributions Based on Negative Binomial Processes

The $$\text {PD}_\alpha ^{(r)}$$ PD α ( r ) distribution, a two-parameter distribution for random vectors on the infinite simplex, generalises the $$\text {PD}_\alpha $$ PD α distribution introduced by Kingman, to which it reduces when $$r=0$$ r = 0 . The parameter $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) arises from its construction based on ratios of ordered jumps of an $$\alpha $$ α -stable subordinator, and the parameter $$r>0$$ r > 0 signifies its connection with an underlying negative binomial process. Herein, it is shown that other distributions on the simplex, including the Poisson–Dirichlet distribution $$\text {PD}(\theta )$$ PD ( θ ) , occur as limiting cases of $$\text {PD}_\alpha ^{(r)}$$ PD α ( r ) , as $$r\rightarrow \infty $$ r → ∞ . As a result, a variety of connections with species and gene sampling models, and many other areas of probability and statistics, are made.