Stationary initialization of a scalar linear state-space model ($S_0={\sum }_{j\ge 0}{\varphi }^j{Z}_{-j}$)

For $|\varphi |<1$, initialize the state by $S_0={\sum }_{j\ge 0}{\varphi }^j{Z}_{-j}$, with variance ${\sigma }_z^2/(1-{\varphi }^2)$. 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.