Quantitative Multidimensional Central Limit Theorems for Means of the Dirichlet-Ferguson Measure

Abstract

. The Dirichlet-Ferguson measure is a cornerstone in nonparametric Bayesian statistics and the study of the distributional properties of expectations with respect to such measure is an important line of research initiated in Cifarelli and Regazzini (1979a,b) and still very active, see Letac and Piccioni (2018) and Lijoi and Prünster (2009). In this paper we provide explicit upper bounds for the d 3 , the d 2 and the convex distances between random vectors whose components are means of the Dirichlet-Ferguson measure and a random vector distributed according to the multivariate Gaussian law. These results are applied to the Gaussian approximation of linear transformations of random vectors with the Dirichlet distribution, yielding presumably optimal rates on the d 3 and the d 2 distances and presumably suboptimal rates on the convex and the Kolmogorov distances.