Asymptotics of predictive distributions driven by sample means and variances

Abstract 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