= Solution
Take the latent autoregression as the scalar state $S_t$. A <state-space model> is
$$
\boxed{\text{state: }S_t=\phi S_{t-1}+Z_t,\qquad\text{observation: }Y_t=S_t+W_t.}
$$
The transition and observation matrices are both scalar, $F=\phi$, $H=1$. 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 is
$$
S_0=\sum_{j\geq0}\phi^jZ_{-j},\qquad\mathbb ES_0=0,\qquad\operatorname{Var}(S_0)=\frac{\sigma_z^2}{1-\phi^2}.
$$
It 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 $S_0\sim N(0,Q/(1-\phi^2))$, 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.
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