# When is a ARMA(p,q) process ergodic?

We know that a ARMA(p,q) process is weakly stationary, iff there is no root of the characteristic polynomial of its AR part lying on the unit circle.

But what is the necessary and sufficient condition for a ARMA(p,q) process to be ergodic? Any book on that?

by "ergodic", I mean its definition in terms of that the first and second moments of the process can be approximated by the sample moments of its single sample path.

• I think that the suggested condition above implies only that the process is ergodic in terms of empirical mean... (?) – user96988 Dec 3 '15 at 13:03

It is sufficient for any stationary process that the autocovariances are absolute summable, i.e. that $\sum_{j=0}^\infty |\gamma_j|<\infty$. You can find this in Hamilton's Time Series Analysis.
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