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

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Abstract

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

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Notes

  1. To simplify notation, we suppress the dependence of the \(\tilde{J}_i\) on r in this section.

  2. \(\mathbf 1\) is a vector in \(\mathbb {R}^\infty \) all of whose components are 1.

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Acknowledgements

We are very grateful to a referee for a very detailed and careful review and suggestions which helped us prepare a better paper.

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Authors and Affiliations

  1. Research School of Finance, Actuarial Studies and Statistics, Australian National University, Canberra, Australia

    Yuguang Ipsen & Ross Maller

  2. School of Mathematics, Statistics and Computer Sciences, University of Tehran, Tehran, Iran

    Soudabeh Shemehsavar

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  1. Yuguang Ipsen
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  2. Ross Maller
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  3. Soudabeh Shemehsavar
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Correspondence to Soudabeh Shemehsavar.

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Ross Maller was partially supported by ARC Grant DP160103037.

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Ipsen, Y., Maller, R. & Shemehsavar, S. Limiting Distributions of Generalised Poisson–Dirichlet Distributions Based on Negative Binomial Processes. J Theor Probab 33, 1974–2000 (2020). https://doi.org/10.1007/s10959-019-00928-7

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  • DOI: https://doi.org/10.1007/s10959-019-00928-7

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