The question relates to a specific exercise:
Let $x_t = a + bt + u_t$, where $u_t \sim IID(0,\sigma^2)$, $b_t \neq 0$ and $0 < \sigma < \infty $. Is $x_t$ a) weakly stationary, b) mean ergodic? Specify the mean and the autocovariance function.
We define weak stationarity as follows:
A time series $x_t$ is said to be weakly stationary if its mean and autocovariance do not depend on the time index, i.e. $E[x_t]=\mu, \forall t$ and $cov(x_t,x_{t-\tau})=\gamma(|\tau|), \forall t,\tau$.
As to mean ergodicity, the following condition is given:
If $\sum \limits_{\tau=0}^{\infty} |\gamma(\tau)| < \infty$, then $x_t$ is mean ergodic.
I proceeded as follows:
- $E[x_t] = a + bt$, which implies the process isn't weakly stationary.
- Let $\tau \geq 0$. Then, $cov(x_t,x_{t-\tau})= E[(x_t-E[x_t])(x_{t-\tau}-E[x_{t-\tau}])]=E[u_t u_{t-\tau}]$. Which leads me to conclude that $\gamma(\tau)=\sigma^2$ if $\tau =0$ and $\gamma(\tau)=0$ if $\tau >0$.
Now, if the results in 2. are correct, the autocovariances should be infinitely summable: $\sum \limits_{\tau=0}^{\infty} |\gamma(\tau)| = \sigma^2 < \infty$. However, the process would then be mean ergodic, which cannot be since mean ergodicity implies weak stationarity.
Where did I go astray??