How do I restrict coefficient values to ensure stationarity and invertibility of ARMA(p,q)? I am trying to simulate an ARMA series but I am concerned about it being conformed to ensure invertibility of $\text{ARMA}(p,q)$ in addition to stationarity. I know of the following conditions for MA series:

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*For an $\text{MA}(1)$ model:  $−1 < \theta_1 < 1$.

*For an $\text{MA}(2)$ model:  $−1 < \theta_2 < 1$, $\theta_2 + \theta_1 > −1$, $ \theta_1 − \theta_2 < 1$.

Moving average models in Forecasting Principles and Practice by Rob J Hyndman and George Athanasopoulos
I also know of the following conditions for AR series:


*For an $\text{AR}(1)$ model:  $−1 < \phi_1 < 1$.

*For an $\text{AR}(2)$ model:  $−1 < \phi_2 < 1$, $\phi_2 + \phi_1 > −1$, $ \phi_1 − \phi_2 < 1$.

Autoregressive models in Forecasting Principles and Practice by Rob J Hyndman and George Athanasopoulos
What conditions holds for $\text{ARMA}(1,1)$ and more?

This is worked out in the book 'Time Series Analysis and Its Applications With R Examples, Third Edition' by Robert Shumway and David Stoffer.
Following their notation, let an $ARMA(p, q)$ model be expressed as $\phi(B)x_t = \theta(B)w_t$ (see equation (3.21) in page 93 of the book), where $\phi(z) = 1 - \phi_1 z − \cdots − \phi_p z^p$ with $\phi_p \neq 0$, $\theta(z) = 1 + \theta_1 z + \cdots + \theta_q z^q$ with $\theta_q \neq 0$, $z$ is a complex number and $B$ is the backshift operator (see Definition 3.6, page 94). Then, an $ARMA(p, q)$ model is invertible if and only if $\theta(z) \neq 0$ for $|z| \le 1$ (see Property 3.2 in page 95). The way of proving this statement is also discussed there.
Also check out these lectures notes. The stationarity of $ARMA(p, q)$ is ensured when all the (real and complex) roots $z_j$ of the polynomial $\phi(z)$ satisfy $|z_j| > 1$ (see page 11 of the lecture notes), i.e., $\phi(z) \neq 0$ for $|z| \le 1$.
In the particular case of $ARMA(1,1)$, $\phi(z) = 1 - \phi_1 z$ and $\theta(z) = 1 + \theta_1 z$. So, it is invertible and stationary if $0 < |\phi_1|, |\theta_1| < 1$.
