The PDα(r)\documentclass[12pt]{minimal}
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\begin{document}$$\text {PD}_\alpha ^{(r)}$$\end{document} distribution, a two-parameter distribution for random vectors on the infinite simplex, generalises the PDα\documentclass[12pt]{minimal}
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\begin{document}$$\text {PD}_\alpha $$\end{document} distribution introduced by Kingman, to which it reduces when r=0\documentclass[12pt]{minimal}
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\begin{document}$$r=0$$\end{document}. The parameter α∈(0,1)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \in (0,1)$$\end{document} arises from its construction based on ratios of ordered jumps of an α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}-stable subordinator, and the parameter r>0\documentclass[12pt]{minimal}
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\begin{document}$$r>0$$\end{document} signifies its connection with an underlying negative binomial process. Herein, it is shown that other distributions on the simplex, including the Poisson–Dirichlet distribution PD(θ)\documentclass[12pt]{minimal}
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\begin{document}$$\text {PD}(\theta )$$\end{document}, occur as limiting cases of PDα(r)\documentclass[12pt]{minimal}
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\begin{document}$$\text {PD}_\alpha ^{(r)}$$\end{document}, as r→∞\documentclass[12pt]{minimal}
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\begin{document}$$r\rightarrow \infty $$\end{document}. As a result, a variety of connections with species and gene sampling models, and many other areas of probability and statistics, are made.
