Asymptotics of predictive distributions driven by sample means and variances

Let alpha Subscript n Baseline left parenthesis dot right parenthesis equals double struck upper P left parenthesis upper X Subscript n plus 1 Baseline element of dot vertical bar upper X 1 comma ellipsis comma upper X Subscript n Baseline right parenthesis α n ( ⋅ ) = P ( X n + 1 ∈ ⋅ ∣ X 1 , … , X n ) $\alpha_n({\cdot})=\mathbb{P}\bigl(X_{n+1}\in{\cdot}\mid X_1,\ldots,X_n\bigr)$ be the predictive distributions of a sequence left parenthesis upper X 1 comma upper X 2 comma ellipsis right parenthesis ( X 1 , X 2 , … ) $(X_1,X_2,\ldots)$ of p -dimensional random vectors. Suppose alpha Subscript n Baseline equals script upper N left parenthesis upper M Subscript n Baseline comma upper Q Subscript n Baseline right parenthesis α n = N ( M n , Q n ) $\alpha_n=\mathcal{N}(M_n,Q_n)$ , where upper M Subscript n Baseline equals left parenthesis 1 divided by n right parenthesis sigma summation Underscript i equals 1 Overscript n Endscripts upper X Subscript i M n = ( 1 / n ) ∑ i = 1 n X i $M_n=({1}/{n})\sum_{i=1}^nX_i$ and upper Q Subscript n Baseline equals left parenthesis 1 divided by n right parenthesis sigma summation Underscript i equals 1 Overscript n Endscripts left parenthesis upper X Subscript i Baseline minus upper M Subscript n Baseline right parenthesis left parenthesis upper X Subscript i Baseline minus upper M Subscript n Baseline right parenthesis Superscript down tack Q n = ( 1 / n ) ∑ i = 1 n ( X i − M n ) ( X i − M n ) ⊤ $Q_n=({1}/{n})\sum_{i=1}^n(X_i-M_n)(X_i-M_n)^{\top}$ . Then there is a random probability measure alpha α $\alpha$ on the Borel subsets of double struck upper R Superscript p R p $\mathbb{R}^p$ such that StartMetric alpha Subscript n Baseline minus alpha EndMetric long right arrow Overscript normal a period normal s period Endscripts 0 ‖ α n − α ‖ ⟶ a . s . 0 $\lVert\alpha_n-\alpha\rVert\overset{\mathrm{a.s.}}\longrightarrow 0$ , where StartMetric dot EndMetric ‖ ⋅ ‖ $\lVert{\cdot}\rVert$ is the total variation distance. An explicit expression for alpha α $\alpha$ is provided and the convergence rate of StartMetric alpha Subscript n Baseline minus alpha EndMetric ‖ α n − α ‖ $\lVert\alpha_n-\alpha\rVert$ is shown to be arbitrarily close to n Superscript negative 1 divided by 2 n − 1 / 2 $n^{-1/2}$ . Moreover, it is still true that StartMetric alpha Subscript n Baseline minus alpha EndMetric long right arrow Overscript normal a period normal s period Endscripts 0 ‖ α n − α ‖ ⟶ a . s . 0 $\lVert\alpha_n-\alpha\rVert\overset{\mathrm{a.s.}}\longrightarrow 0$ even if alpha Subscript n Baseline equals script upper L left parenthesis upper M Subscript n Baseline comma upper Q Subscript n Baseline right parenthesis α n = L ( M n , Q n ) $\alpha_n=\mathcal{L}(M_n,Q_n)$ , where script upper L L $\mathcal{L}$ belongs to a class of distributions much larger than the normal. The predictives alpha Subscript n α n $\alpha_n$ are useful in various frameworks, including Bayesian predictive inference and predictive resampling. Finally, the asymptotic behavior of copula-based predictive distributions is investigated and a numerical experiment is performed.