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Limiting Behaviour of Poisson-Dirichlet and Generalised Poisson-Dirichlet Distributions

Abstract

We derive large-sample and other limiting distributions of the ``frequency of frequencies'' vector, Mn{\bf M_n}, together with the number of species, KnK_n, in a Poisson-Dirichlet or generalised Poisson-Dirichlet gene or species sampling model. Models analysed include those constructed from gamma and α\alpha-stable subordinators by Kingman, the two-parameter extension by Pitman and Yor, and another two-parameter version constructed by omitting large jumps from an α\alpha-stable subordinator. In the Poisson-Dirichlet case Mn{\bf M_n} and KnK_n turn out to be asymptotically independent, and notable, especially for statistical applications, is that in other cases the conditional limiting distribution of Mn{\bf M_n}, given KnK_n, is normal, after certain centering and norming.

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