= Finite-past innovation
{title2=$U_t=X_t-\operatorname{proj}_{\operatorname{span}(X_1,\ldots,X_{t-1})}X_t$}
Starting observations at a finite time, subtract the best linear predictor based on the available earlier observations. The resulting residual is orthogonal to their linear span. The unit-triangular relation between observations and residuals makes the residuals an <orthogonal basis> of the same finite observation space. These differ from infinite-past innovations until the initialization effect disappears.
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