A Berry-Esseen theorem for Pitman's $\alpha$-diversity

TL;DR

This paper establishes a Berry-Esseen type theorem quantifying the convergence rate of the scaled number of distinct elements in a sample from a Poisson-Dirichlet distribution to its limiting scaled Mittag-Leffler distribution.

Contribution

It provides the first explicit bound on the convergence rate in the CLT for the number of distinct elements under the Poisson-Dirichlet model, using novel probabilistic representations and bounds.

Findings

Convergence rate of order 1/n^α for the scaled number of distinct elements.

Explicit constant C(α, θ) for the bound.

New probabilistic representation of K_n as a compound distribution.

Abstract

This paper is concerned with the study of the random variable $K_n$ denoting the number of distinct elements in a random sample $(X_1, \dots, X_n)$ of exchangeable random variables driven by the two parameter Poisson-Dirichlet distribution, $PD(\alpha,\theta)$. For $\alpha\in(0,1)$, Theorem 3.8 in \cite{Pit(06)} shows that $\frac{K_n}{n^{\alpha}}\stackrel{\text{a.s.}}{\longrightarrow} S_{\alpha,\theta}$ as $n\rightarrow+\infty$. Here, $S_{\alpha,\theta}$ is a random variable distributed according to the so-called scaled Mittag-Leffler distribution. Our main result states that $$ \sup_{x \geq 0} \Big| \ppsf\Big[\frac{K_n}{n^{\alpha}} \leq x \Big] - \ppsf[S_{\alpha,\theta} \leq x] \Big| \leq \frac{C(\alpha, \theta)}{n^{\alpha}} $$ holds with an explicit constant $C(\alpha, \theta)$. The key ingredients of the proof are a novel probabilistic representation of $K_n$ as compound distribution and new, refined versions of certain quantitative bounds for the Poisson approximation and the compound Poisson distribution.