= Stationary initialization of an ARMA(1,1) state
{title2=$P=\begin{pmatrix}\gamma_0&0&\sigma^2\\0&\sigma^2&0\\\sigma^2&0&\sigma^2\end{pmatrix}$}
For $|\phi|<1$ and centered <independent> <white noise> values, the state above has <expected value> zero and this stationary <covariance matrix>, where $\gamma_0=\sigma^2(1+\theta^2+2\phi\theta)/(1-\phi^2)$. The cross-covariance with the future <white noise> value is zero and that with the contemporaneous <white noise> value is $\sigma^2$. Direct multiplication verifies $P=GPG^\top+Q$.
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