# condition for a ARMA process to be wide-sense stationary

For a ARMA process,

• some (e.g. in Tsay's Financial Time Series) said: it is wide-sense stationary, iff all the roots of its AR characteristic polynomial are greater than 1 in magnitude. This is shown by requiring the variance of the series at any time to be nonnegative and constant.

• some (e.g. in Brockwell's Introduction to Time Series) said: it is wide-sense stationary, iff all the roots of its AR characteristic polynomial are not equal to 1 in magnitude. It is causal, iff all the roots of its AR characteristic polynomial are greater than 1 in magnitude.

Questions

1. According to Brockwell, it seems that a ARMA process can be wide-sense stationary while non-causal, iff all the roots of its AR characteristic polynomial are less than 1 in magnitude.

For example, in AR(1) model $X_t = a X_{t-1} + e_t$, the root of its AR characteristic polynomial is less than 1 in magnitude, iff $|a|<1$.

I wonder what $Var(X_t)$ in this AR(1) model is then? Is it still well defined (nonnegative)?

How does its autocorrelation function behave as the lag increases? (not decay, but stay at 1?)

2. I wonder how you understand the difference in the necessary and sufficient conditions fpr wide-sense stationarity given by Tsay and by Brockwell?

Thanks!