Let $α_n(\cdot)=P\bigl(X_{n+1}\in\cdot\mid X_1,\ldots,X_n\bigr)$ be the predictive distributions of a sequence $(X_1,X_2,\ldots)$ of $p$-dimensional random vectors. Suppose $$α_n= \mathcal{N} _p (M_n,Q_n)$$ 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 $α$ on the Borel subsets of $\mathbb{R}^p$ such that $\lVertα_n-α\rVert\overset{a.s.}\longrightarrow 0$ where $\lVert\cdot\rVert$ is total variation distance. An explicit expression for $α$ is provided and the convergence rate of $\lVertα_n-α\rVert$ is shown to be arbitrarily close to $n^{-1/2}$. Moreover, it is still true that $\lVertα_n-α\rVert\overset{a.s.}\longrightarrow 0$ even if $α_n=\mathcal{L}(M_n,Q_n)$ where $\mathcal{L}$ belongs to a class of distributions much larger than the normal. The predictives $α_n$ 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.