Asymptotics of predictive distributions driven by sample means and variances

Abstract

Let \(\alpha_n(\cdot)=P\bigl(X_{n+1}\in\cdot\mid X_1,\ldots,X_n\bigr)\) be the predictive distributions of a sequence \((X_1,X_2,\ldots)\) of \(p\)-variate random variables. Suppose \[\alpha_n=\mathcal{N}_p(M_n,Q_n)\] where \(M_n=\frac{1}{n}\sum_{i=1}^nX_i\) and \(Q_n=\frac{1}{n}\sum_{i=1}^n(X_i-M_n)(X_i-M_n)^t\). Then, there is a random probability measure \(\alpha\) on \(\mathbb{R}^p\) such that \(\alpha_n\rightarrow\alpha\) weakly a.s. If \(p\in\{1,2\}\), one also obtains \(\lVert\alpha_n-\alpha\rVert\overset{a.s.}\longrightarrow 0\) where \(\lVert\cdot\rVert\) is total variation distance. Moreover, the convergence rate of \(\lVert\alpha_n-\alpha\rVert\) is arbitrarily close to \(n^{-1/2}\). These results (apart from the one regarding the convergence rate) still apply even if \(\alpha_n=\mathcal{L}_p(M_n,Q_n)\), where \(\mathcal{L}_p\) belongs to a class of distributions much larger than the normal. Finally, the asymptotic behavior of copula-based predictive distributions (introduced in [13]) is investigated and a numerical experiment is performed.