A Berry-Esseen theorem for Pitman's $α$-diversity

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, $PD(α,θ)$. For $α\in(0,1)$, Theorem 3.8 in \cite{Pit(06)} shows that $\frac{K_n}{n^α}\stackrel{\text{a.s.}}{\longrightarrow} S_{α,θ}$ as $n\rightarrow+\infty$. Here, $S_{α,θ}$ is a random variable distributed according to the so-called scaled Mittag-Leffler distribution. Our main result states that $$ \sup_{x \geq 0} \Big| \ppsf\Big[\frac{K_n}{n^α} \leq x \Big] - \ppsf[S_{α,θ} \leq x] \Big| \leq \frac{C(α, θ)}{n^α} $$ holds with an explicit constant $C(α, θ)$. 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.