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

Abstract

This paper contributes to the study of the random number $K_{n}$ of blocks in the random partition of $\{1,\ldots,n\}$ induced by random sampling from the celebrated two parameter Poisson–Dirichlet process. For any $\alpha \in (0,1)$ and $\theta >-\alpha $ Pitman (Combinatorial Stochastic Processes (2006) Springer, Berlin) showed that $n^{-\alpha }K_{n}\stackrel{\text{a.s.}}{\longrightarrow }S_{\alpha,\theta }$ as $n\rightarrow +\infty $, where the limiting random variable, referred to as Pitman’s $\alpha $-diversity, is distributed according to a polynomially scaled Mittag–Leffler distribution function. Our main result is a Berry–Esseen theorem for Pitman’s $\alpha $-diversity $S_{\alpha,\theta }$, namely we show that \[\mathop{\mathrm{sup}}_{x\geq 0}\biggl\vert \mathsf{P}\biggl[\frac{K_{n}}{n^{\alpha }}\leq x\biggr]-\mathsf{P}[S_{\alpha,\theta }\leq x]\biggr\vert \leq\frac{C(\alpha,\theta )}{n^{\alpha }}\] holds for every $n\in \mathbb{N}$ with an explicit constant term $C(\alpha,\theta )$, for $\alpha \in (0,1)$ and $\theta >0$. The proof relies on three intermediate novel results which are of independent interest: (i) a probabilistic representation of the distribution of $K_{n}$ in terms of a compound distribution; (ii) a quantitative version of the Laplace’s approximation method for integrals; (iii) a refined quantitative bound for Poisson approximation. An application of our Berry–Esseen theorem is presented in the context of Bayesian nonparametric inference for species sampling problems, quantifying the error of a posterior approximation that has been extensively applied to infer the number of unseen species in a population.