Uniform rates of the Glivenko-Cantelli convergence and their use in approximating Bayesian inferences

Abstract

This paper deals with the problem of quantifying the approximation a probability measure by means of an empirical (in a wide sense) random probability measure, depending on the first n terms of a sequence of random elements. In Section 2, one studies the range of oscillation near zero of the Wasserstein distance $\ud^{(p)}_{\pms}$ between $\pfrak_0$ and $\hat{\pfrak}_n$, assuming that the $\xitil_i$'s are i.i.d. with $\pfrak_0$ as common law. Theorem 2.3 deals with the case in which $\pfrak_0$ is fixed as a generic element of the space of all probability measures on $(\rd, \mathscr{B}(\rd))$ and $\hat{\pfrak}_n$ coincides with the empirical measure. In Theorem 2.4 (Theorem 2.5, respectively) \pfrak_0 is a d-dimensional Gaussian distribution (an element of a distinguished type of statistical exponential family, respectively) and $\hat{\pfrak}_n$ is another $d$-dimensional Gaussian distribution with estimated mean and covariance matrix (another element of the same family with an estimated parameter, respectively). These new results improve on allied recent works (see, e.g., [31]) since they also provide uniform bounds with respect to $n$, meaning that the finiteness of the p-moment of the random variable $\sup_{n \geq 1} b_n \ud^{(p)}_{\pms}(\pfrak_0, \hat{\pfrak}_n)$ is proved for some suitable diverging sequence b_n of positive numbers. In Section 3, under the hypothesis that the $\xitil_i$'s are exchangeable, one studies the range of the random oscillation near zero of the Wasserstein distance between the conditional distribution--also called posterior--of the directing measure of the sequence, given $\xitil_1, \dots, \xitil_n$, and the point mass at $\hat{\pfrak}_n$. In a similar vein, a bound for the approximation of predictive distributions is given. Finally, Theorems from 3.3 to 3.5 reconsider Theorems from 2.3 to 2.5, respectively, according to a Bayesian perspective.