Weak Stationary Time Series If $\{X_t : t \in Z\}$ is a weakly stationary time series does that necessarily mean that  $\{X^2_t : t \in Z\}$ weakly stationary time series as well? I can't really think of an example to prove or disprove this statement.

No, it does not. Weak stationarity requires finite variance. So let $X_t$ have finite variance, but not finite 3th moment. Then $X_t^2$ does not have finite second moment,i.e. variance does not exist, therefore can not be weakly stationary.
