ARMA Filter Output Stationary and set up?
Hi: The use of the term &quot;stationary&quot; is vague because there are a few different kinds. But usually, the term &quot;stationary&quot; is meant to mean wide-sense stationary. But, since you've also used the term wide sense stationary in your question, I'm thinking that you're use of the term stationary might just be referring to the output ( by output, I'm assuming you mean the estimated residual noise term )  having a constant (zero) mean and constant variance.
Wide sense stationary refers to constant mean, constant variance and $corr(X_{t1}, X_{t2})$ being only a function of, the time difference, $t2 - t1$.
But, the general process in ARMA modelling ( in statistical time series that is. dsp could be different), is to take the estimated residuals and check that they have a  constant mean of zero, constant variance and that they are uncorrelated. This is because the ARMA framework assumes that the error term is white noise so the estimated residuals should be uncorrelated and have zero mean and constant variance.
So, the test that is done after the ARMA model is identified and estimated
is testing that the estimated residuals are white noise. But, white noise is a specific case of stationarity so one could argue that you're kind of testing for wide-sense stationarity ( except that corr should be zero so not even a function of the time difference ).
Note that there is another term called &quot;strict sense stationarity&quot; which is a stronger form of stationarity where one requires that the joint distribution of  $X_{1}, \ldots, X_{n}$ is the same as the joint distribution of  $X_{t+1}, \ldots, X_{t+n}$.