Examples of Covariance Stationary Time Series I seem to be having trouble in comprehending what it means for a time series to be covariance stationary. Specifically, with the third condition that for any $t,h$ the $cov(x_t,x_{t+h})$ 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?
Let $\epsilon_1, \epsilon_2, ... ,\epsilon_t,...$ denote i.i.d. stochastic variables, each having a standard deviation of $\sigma$.

Let
$$
x_t = \sum_{i = 1}^t \epsilon_i.
$$
Then
$$
cov(x_t,x_{t+h}) =
cov\left(
\sum_{i = 1}^t \epsilon_i,
\sum_{i = 1}^{t+h} \epsilon_i
\right).
$$
Because of the i.i.d. property
$$
cov\left(
\sum_{i = 1}^t \epsilon_i,
\sum_{i = 1}^{t+h} \epsilon_i
\right)
=
\sum_{i = 1}^t cov\left(\epsilon_i,\epsilon_i\right) +
\sum_{i = 1}^t\sum_{j \neq i} cov\left(\epsilon_i,\epsilon_j\right)
$$
and furthermore because of independence
$$
\sum_{i = 1}^t cov\left(\epsilon_i,\epsilon_i\right) +
\sum_{i = 1}^t\sum_{j \neq i} cov\left(\epsilon_i,\epsilon_j\right)
=
\sum_{i = 1}^t \sigma +
\sum_{i = 1}^t\sum_{j \neq i} 0
=
t \cdot \sigma.
$$
Now we have shown that for this time series
$$
cov(x_t,x_{t+h}) =
t \cdot \sigma
$$
depends on $t$ hence it is not covariance stationary.

For a covariance stationary time series, let
$$
z_t = \epsilon_{t-1} + \epsilon_{t}.
$$
By performing the same analyis, you can conlude that for $t &gt; 1$,
$$
cov(z_t,z_{t+h}) =
\left\{
\begin{array}{ll}
\sigma &amp; \text{ if } h = 1 \\
0 &amp; \text{ if } h &gt; 1.
\end{array}
\right.
$$
Since this depends on $h$, but not on $t$, the time series $z_t$ will be covariance stationary (after $t = 1$).