Best linear forecast from a finite history

Let $X=(X_t{)}_{t\in \mathbb{Z}}$ be a weakly stationary real process on one probability space, with common mean $\mu$, and let $n,h\ge 1$ be integers. Its finite-history best linear forecast of $X_{n+h}$ from $X_1,\dots ,X_n$ is the unique almost-sure class minimizing squared $L^2$ error over the affine subspace $M_n:=\{[a_0+a_1{X}_1+\cdots +a_n{X}_n]:a_0,\dots ,a_n\in \mathbb{R}\}\subseteq L^2(\mathbb{P})$. Here $[Z]$ denotes the almost-sure equivalence class of a square-integrable random variable, as in The space $L^p(\mu )$ as the quotient by null functions, and the finite linear span uses Linear combination of a finite list, and the span $\mathrm{span}(S)$ as the smallest linear subspace containing $S$. Weak stationarity and its moment conditions are those of Strict and weak stationarity for integer-indexed time series. Existence and uniqueness of the minimizing class are established by Existence and uniqueness of the finite-history predictor class ↗. The term best linear refers to this affine $L^2$ projection; it does not assert equality with a conditional expectation without additional hypotheses.