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

TL;DR

This paper investigates the asymptotic behavior of the frequency of frequencies vector and the number of species in various Poisson-Dirichlet models, revealing independence and normality properties useful for statistical applications.

Contribution

It derives new large-sample and limiting distributions for key statistics in generalized Poisson-Dirichlet models, including models from gamma and stable subordinators.

Findings

${f M_n}$ and $K_n$ are asymptotically independent in the Poisson-Dirichlet case.

Conditional distribution of ${f M_n}$ given $K_n$ is asymptotically normal.

Results apply to models constructed from gamma and $ ext{alpha}$-stable subordinators.

Abstract

We derive large-sample and other limiting distributions of the ``frequency of frequencies'' vector, ${\bf M_n}$, 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 $\alpha$-stable subordinators by Kingman, the two-parameter extension by Pitman and Yor, and another two-parameter version constructed by omitting large jumps from an $\alpha$-stable subordinator. In the Poisson-Dirichlet case ${\bf M_n}$ 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 ${\bf M_n}$, given $K_n$, is normal, after certain centering and norming.