Let $X\{(n)\}_{n\in \mathbb Z}$ be a weakly stationary stochastic process.
Given the information up to time $n$, $\{X_k\vert k\leq n\}$, in what way does the optimal linear predition change when I ask for the best linear predictor for $X(n+l),l>1$ instead of $X(n+1)$?
Intuitively, they should be the same, since I don't gain any information, I just try to look further into the future.
The optimal predictor depends on the prediction length $l$. To see this, define the prediction error
$$\epsilon_k = X_k - \sum_{m=0}^{\infty}w_m X_{k-l-m}$$
where $w_k$ are the coefficients of the prediction filter. If you use the common MSE criterion you get
$$E\{\epsilon_k^2\} \rightarrow \min_{w_k}$$
where $E\{\cdot\}$ denotes the expectation operator. This leads to the orthogonality condition
$$E\{\epsilon_kX_{k-l-n}\}=0 \text{ for } n\ge 0 $$
which is equivalent to
$$\sum_{m=0}^{\infty}w_mr_X(n-m)=r_X(n+l)\text{ for } n\ge 0 \tag{1}$$
where $r_X(n)$ is the autocorrelation function of $X_k$. From (1), the optimal coefficients $w_k$ can be computed. Note that they depend on the prediction length $l$.