A BERRY-ESSEEN THEOREM FOR PITMAN'S α-DIVERSITY

This paper contributes to the study of the random number K-n of blocks in the random partition of {1, ..., n} induced by random sampling from the celebrated two parameter Poisson-Dirichlet process. For any alpha is an element of (0, 1) and theta > -alpha Pitman (Combinatorial Stochastic Processes (2006) Springer, Berlin) showed that n(-alpha) K-n ->(a.s.) S-alpha,S-theta as n -> +infinity, 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 sup(x >= 0)vertical bar P[K-n/n(alpha) <= x] - P[S-alpha,S-theta <= x]vertical bar <= C(alpha,theta)/n(alpha) holds for every n is an element of N with an explicit constant term C(alpha, theta), for alpha is an element of (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.