Asymptotics of predictive distributions driven by sample means and variances

Let αn(·) = P(Xn+1 ∈ · | X1, . . . ,Xn) be the predictive distributions of a sequence (X1,X2, . . .) of p-dimensional random vectors. Suppose αn = N(Mn,Qn) where Mn is the sample mean and Qn the sample covariance matrix of (X1,...,Xn). Then, there is a random probability measure α on the Borel subsets of Rp such that ∥αn − α∥ a.s. −→ 0 where ∥·∥ is total variation distance. An explicit expression for α is provided and the convergence rate of ∥αn − α∥ is shown to be arbitrarily close to n^−1/2. Moreover, it is still true that ∥αn − α∥ a.s. −→ 0 even if αn = L(Mn,Qn) where L belongs to a class of distributions much larger than the normal. The predictives αn are useful in various frameworks, including Bayesian predictive inference and predictive resampling. Finally, the asymptotic behavior of copula-based predictive distributions (introduced in [20]) is investigated and a numerical experiment is performed.