Abstract
The PD alpha(r)distribution, a two-parameter distribution for random vectors on the infinite simplex, generalises the PD alpha distribution introduced by Kingman, to which it reduces when r=0 The parameter alpha is an element of(0,1) arises from its construction based on ratios of ordered jumps of an alpha-stable subordinator, and the parameter r0signifies its connection with an underlying negative binomial process. Herein, it is shown that other distributions on the simplex, including the Poisson-Dirichlet distribution PD(theta)occur as limiting cases of PD alpha(r) as r ->infinity As a result, a variety of connections with species and gene sampling models, and many other areas of probability and statistics, are made.