Show that linear predictor of a stationary Gaussian time series depends only on the first 2 moments of the series I would like to convince myself that the best predictor of a stationary Gaussian time series depends on more than just the first 2 unconditional moments of the series, whereas the best linear predictor depends only on the first 2 moments. Let "best" be defined, as usual, in terms of minimal mean squared error.
"... the fact that the best linear predictor depends only on the mean and the ACF of the series means that it can be calculated without more detailed knowledge of the joint distribution" Brockwell & Davis page 40.

Suppose $x_t$ follows a stationary AR(1) process,
$$
x_t=\varphi x_{t-1}+\varepsilon_t.
$$
An infeasible optimal* forecast of $x_{t+1}$ is $\hat{x}_{t+1|t}=\varphi x_t$. If we can estimate $\varphi$ with sufficient precision (which we often can, given a sufficiently large sample), this forecast will be better than a forecast that only depends on $\bar{x}$ and $\hat\sigma$, since $\varphi x_t$ cannot be expressed by them using a formula that is independent of $t$.
*Optimal under square loss = the one that minimizes the expected square error.
Update: this answer applies to my interpretation of the original post before clarification.
