= Stationary initialization of a scalar linear state-space model
{title2=$S_0=\sum_{j\geq0}\phi^jZ_{-j}$}
For $|\phi|<1$, initialize the state by $S_0=\sum_{j\geq0}\phi^jZ_{-j}$, with variance $\sigma_z^2/(1-\phi^2)$. It is orthogonal to future state noise and to observation noise orthogonal to all state noise. Under a Gaussian specification, take the initial state Gaussian with this variance and independent of future noises. Starting at zero instead gives a transient model.
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