Let $z_{t}$ be ARMA(1,1) process. $$ z_{t+1} = \phi z_{t} + \theta\varepsilon_{t} + \varepsilon_{t+1} $$
In order to have a stationary process we must have $|\phi| < 1$. This is clear. The auto-correlation function is for $\theta : = \theta$ and $\theta : = \frac{1}{\theta}$ is unchangeable, as discussed here ARMA model with MA coefficient greater than 1
Though, the auto-covariance function (and, consequently, the variance) does depend on $\theta$, i.e., for example for $\tau =0, 1$: $$ \gamma(0) = \sigma^2 \frac{1 + 2\phi\theta + \theta^2}{1 - \phi^2} $$ and $$ \gamma(1) = \sigma^2 \frac{(\phi + \theta)(1 + \phi\theta)}{1 - \phi^2} $$
The question: why do we use then condition $|\theta| < 1$? Do we need it in some computational algorithms which use the auto-correlation function?
