Predictive Distribution of Time series with Uncertain Future Values

Question

Lets say that we have $\{x_1, \cdots, x_n\}$ as a time series, probably a Gaussian one with $f_{X_1, \cdots, X_n}(x_1, \cdots, x_n)$ as its joint distribution function. We want to estimate the predictive distribution of the next $k$ uncertain samples, having the previous $n$ samples, i.e., we are looking to find the $f_{X_{n+1}, \cdots, X_{n+k}}(x_{n+1}, \cdots, x_{n+k} | x_1, \cdots, x_n)$. With the Normal distribution assumption, what would be the mean and variance of the predictive distribution and how could we estimate them using known $n$ samples.