Asymptotics of predictive distributions driven by sample means and variances

Let αn(·)=Pbigl(Xn+1∈·mid X1,ldots,Xnbigr) be the predictive distributions of a sequence (X1,X2,ldots) of p-dimensional random vectors. Suppose αn= N p (Mn,Qn) where Mn=frac1nsumi=1nXi and Qn=frac1nsumi=1n(Xi-Mn)(Xi-Mn)t. Then, there is a random probability measure α on the Borel subsets of ℝp such that lVertαn-αrVertoverseta.s.longrightarrow 0 where lVert·rVert is total variation distance. An explicit expression for α is provided and the convergence rate of lVertαn-αrVert is shown to be arbitrarily close to n-1/2. Moreover, it is still true that lVertαn-αrVertoverseta.s.longrightarrow 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 [13]) is investigated and a numerical experiment is performed.