 Solution

Take the latent autoregression as the scalar state $S_t$. A state-space model isThe transition and observation matrices are both scalar, $F=\varphi$, $H=1$. State-noise variance is $Q={\sigma }_z^2$, observation-noise variance is $R={\sigma }_w^2$, and the cross-noise covariance is zero at every pair of times.

A complete stationary initialization isIt is orthogonal to future state noise and to all observation noise. This is the stationary initialization of a scalar linear state-space model. This specifies the initial state in terms of the actual given two-sided white noise sequence, as well as its second-order law; simply starting from zero would give transient rather than stationary observations.

If a Gaussian state-space specification is intended, the complete specialization is $S_0\sim N(0,Q/(1-{\varphi }^2))$, independent of the future iid Gaussian state and observation noises, themselves independent with variances $Q,R$. Under the printed assumptions alone, Gaussian distributions and independence cannot be deduced from white noise orthogonality; the equations and stationary-series initialization above give the exact second-order representation without adding them.