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, , together with the number of species, , in a Poisson-Dirichlet or generalised Poisson-Dirichlet gene or species sampling model. Models analysed include those constructed from gamma and -stable subordinators by Kingman, the two-parameter extension by Pitman and Yor, and another two-parameter version constructed by omitting large jumps from an -stable subordinator. In the Poisson-Dirichlet case and turn out to be asymptotically independent, and notable, especially for statistical applications, is that in other cases the conditional limiting distribution of , given , is normal, after certain centering and norming.
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