= Recursive forecasts of an autoregressive process
{title2=$\widehat X_{T,k}=m+\sum_{j=1}^p\phi_j(\widehat X_{T,k-j}-m)$}
For a causal <autoregressive process> with mean $m$ and innovations orthogonal to the observed past, define $\widehat X_{T,h}=X_{T+h}$ for $h\le0$ and replace future innovations by zero. The displayed recursion is its <best linear prediction from a finite past> when at least the last $p$ observations are available. Future innovation sums are orthogonal to those observations, which proves the projection property. Mere <white noise> does not guarantee that a linear forecast equals a conditional mean: that stronger interpretation requires martingale-difference or independent innovations.
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