A Berry.Esseen theorem for Pitman.s . . -diversity

This paper contributes to the study of the random number Kn of blocks in the random partition of {1,.,n} induced by random sampling from the celebrated two parameter Poisson.Dirichlet process. For any ..(0,1) and .>.. Pitman (Combinatorial Stochastic Processes (2006) Springer, Berlin) showed that n..Kn.a.s.S.,. as n.+., where the limiting random variable, referred to as Pitman.s .-diversity, is distributed according to a polynomially scaled Mittag.Leffler distribution function. Our main result is a Berry.Esseen theorem for Pitman.s .-diversity S.,., namely we show that supx.0.....[Knn..x]...[S.,..x]....C(.,.)n. holds for every n.. with an explicit constant term C(.,.), for ..(0,1) and .>0. The proof relies on three intermediate novel results which are of independent interest: (i) a probabilistic representation of the distribution of Kn 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.