# Roots of lag polynomial and stationarity

I encountered the following theorem:

If all the roots of the AR lag polynomial of an ARMA process lie outside the unit circle, the process is stationary.

I noticed that this is only an implication. So, if I understood things correctly, we can make this inference:

• If the process is not stationary, then the roots are not all outside the unit circle, i.e. there is at least one root inside the unit circle

However, suppose we are considering a process, for which we found one root inside the unit circle, the theorem does not say that it is not stationary, it could be stationary. Am I correct?

• Your inference is not correct, because there are also other ways to violate stationarity in addition to unit roots, e.g. by nonstationary variance processes. – Richard Hardy Apr 8 at 7:06