Stationary initialization of an ARMA(1,1) state ($P=\left(\begin{array}{ccc}{\gamma }_0& 0& {\sigma }^2\\ 0& {\sigma }^2& 0\\ {\sigma }^2& 0& {\sigma }^2\end{array}\right)$)

For $|\varphi |<1$ and centered independent white noise values, the state above has expected value zero and this stationary covariance matrix, where ${\gamma }_0={\sigma }^2(1+{\theta }^2+2\varphi \theta )/(1-{\varphi }^2)$. 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 $P=GP{G}^{\top }+Q$.