= Best linear prediction from an infinite past
{title2=$\widehat X_{t+1}=\operatorname{proj}_{\mathcal H_t}X_{t+1}$}
The best mean-square linear predictor from a semi-infinite past is the <orthogonal projection> onto the closed linear span of past observations in the random-variable <Hilbert space>. For a causal invertible <ARMA> representation, past observations and past innovations generate the same closed span. The next <linear innovation> is orthogonal to it and is therefore the prediction error.
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