Recursive forecasts of an autoregressive process (${\hat{X}}_{T,k}=m+{\sum }_{j=1}^p{\varphi }_j({\hat{X}}_{T,k-j}-m)$)

For a causal autoregressive process with mean $m$ and innovations orthogonal to the observed past, define ${\hat{X}}_{T,h}=X_{T+h}$ for $h\le 0$ 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.

If the increments of $W$ have mean $\mu$ and centered increments satisfy a causal autoregressive process of order one with $|\varphi |<1$, sum their recursive forecasts to obtain the displayed expression. For fixed forecast origin, ${\hat{W}}_{T,k}/k\to \mu$. One must observe the last increment, requiring $T\ge 2$ or a known initial level $W_0$; a single observed level without an initial-level model does not identify that increment.