I seem to be having trouble in comprehending what it means for a time series to be covariance stationary. Specifically, with the third condition of for any $t$ and $h$, $ cov(x_t,x_{t+h})$ only depends on $h$ and not $t$.
Would anyone have any examples of how a time series might be covariance stationary or any examples of non-covariance stationary time series?
I would really appreciate it!
Thanks.
Example of an covariance stationary time series: Define $X_t=a_ta_{t-1}$ a process with $\{a_t\}$ white noise ($N(0,\sigma^2)$):
\begin{eqnarray*} cov(X_{t},X_{t+h})& =& cov(a_{t}a_{t-1},a_{t+h}a_{t+h-1})\\ &=& E[a_{t}a_{t-1}a_{t+h}a_{t+h-1}]-E[a_{t}a_{t-1}]E[a_{t+h}a_{t+h-1}] \cdots (*)\\ \end{eqnarray*}
where (by distributional assumption $N(0,\sigma^2$): $$cov(a_t,a_{t-1})=0=E[a_ta_{t-1}]-E[a_t]E[a_{t-1}]=E[a_ta_{t-1}]-[0][0]$$ $$E[a_ta_{t-1}]=0$$
substituting the last result in $(*)$: \begin{eqnarray*} cov(X_{t},X_{t+h})&=& E[a_{t}a_{t-1}a_{t+h}a_{t+h-1}]-[0][0] \\ cov(X_{t},X_{t+h})&=& E[a_{t}a_{t-1}a_{t+h}a_{t+h-1}] \end{eqnarray*}
case 1 ($h=0$): \begin{eqnarray*} cov(X_{t},X_{t+h})&=& E[a_{t}a_{t-1}a_{t}a_{t-1}] \\ cov(X_{t},X_{t+h})&=& E[a_{t}^2a_{t-1}^2]=E[a_{t}^2]E[a_{t-1}^2]=\sigma^2*\sigma^2=\sigma^4. \end{eqnarray*}
case 2 ($h\neq 0$): \begin{eqnarray*} cov(X_{t},X_{t+h})&=& E[a_{t}a_{t-1}a_{t+h}a_{t+h-1}] \\ \text{by independence:}\\ cov(X_{t},X_{t+h})&=& E[a_{t}]E[a_{t-1}]E[a_{t+h}]E[a_{t+h-1}]=0*0*0*0=0. \end{eqnarray*}
$\rightarrow Cov(X_t,X_{t+h})=\sigma^4*\mathbb{1}_{\{h=0\}}$ so, the covariance funcion only depends on $h$ and not $t$.