Large deviations associated with Poisson--Dirichlet distribution and Ewens sampling formula

TL;DR

This paper investigates large deviation principles for distributions related to the Poisson--Dirichlet distribution and Ewens sampling formula as the parameter $ heta$ approaches infinity, linking these to models of infinite alleles and population genetics.

Contribution

It provides new large deviation results connecting the finite allele approximation with the Poisson--Dirichlet distribution and compares sample size with effective population size in genetic models.

Findings

Large deviations are characterized for the Poisson--Dirichlet distribution as $ heta$ approaches infinity.

The finite allele approximation is shown to be related to the Poisson--Dirichlet distribution through large deviations.

A comparison between sample size and effective population size is established based on the Ewens sampling formula.

Abstract

Several results of large deviations are obtained for distributions that are associated with the Poisson--Dirichlet distribution and the Ewens sampling formula when the parameter $\theta$ approaches infinity. The motivation for these results comes from a desire of understanding the exact meaning of $\theta$ going to infinity. In terms of the law of large numbers and the central limit theorem, the limiting procedure of $\theta$ going to infinity in a Poisson--Dirichlet distribution corresponds to a finite allele model where the mutation rate per individual is fixed and the number of alleles going to infinity. We call this the finite allele approximation. The first main result of this article is concerned with the relation between this finite allele approximation and the Poisson--Dirichlet distribution in terms of large deviations. Large $\theta$ can also be viewed as a limiting procedure of the effective population size going to infinity. In the second result a comparison is done between the sample size and the effective population size based on the Ewens sampling formula.