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:

*

*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?

I found a very useful material that answers this question the way I expected. Lecture 2: ARMA(p,q) models(part 3)
On page 6 precisely, it goes as follows:

*

*The (stability) stationarity condition is the one of an AR(1) process (or
ARMA(1,0) process) :
$|\phi| < 1$.

*The invertibility condition is the one of a MA(1) process (or
ARMA(0,1) process) :
$|\theta| < 1$.

*The representation of an ARMA(1,1) process is fundamental or causal
if :
$|\phi| < 1$ and $|\theta| < 1$.

*The representation of an ARMA(1,1) process is said to be minimal and
causal if :
$|\phi| < 1$, $|\theta| < 1$ and $\phi \neq \theta$.

