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

09/02/2023

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by   Ross Maller, et al.

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We derive large-sample and other limiting distributions of the “frequency of frequencies” vector, M_n, together with the number of species, K_n, in a Poisson-Dirichlet or generalised Poisson-Dirichlet gene or species sampling model. Models analysed include those constructed from gamma and α-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-stable subordinator. In the Poisson-Dirichlet case M_n and K_n turn out to be asymptotically independent, and notable, especially for statistical applications, is that in other cases the conditional limiting distribution of M_n, given K_n, is normal, after certain centering and norming.