Asymptotic Results for the Two-parameter Poisson-Dirichlet Distribution

TL;DR

This paper establishes moderate and large deviation principles for the two-parameter Poisson-Dirichlet distribution, analyzing its asymptotic behavior as parameters vary, which enhances understanding of its probabilistic properties.

Contribution

It provides the first comprehensive asymptotic analysis of the two-parameter Poisson-Dirichlet distribution under various parameter regimes.

Findings

Moderate deviation principles when θ approaches infinity.

Large deviation principles when α and θ approach zero.

Insights into the distribution's asymptotic behavior.

Abstract

The two-parameter Poisson-Dirichlet distribution is the law of a sequence of decreasing nonnegative random variables with total sum one. It can be constructed from stable and Gamma subordinators with the two-parameters, $\alpha$ and $\theta$, corresponding to the stable component and Gamma component respectively. The moderate deviation principles are established for the two-parameter Poisson-Dirichlet distribution and the corresponding homozygosity when $\theta$ approaches infinity, and the large deviation principle is established for the two-parameter Poisson-Dirichlet distribution when both $\alpha$ and $\theta$ approach zero.