Let $\{Y_t\}$ be the three point moving average process that is defined as $\{Y_t\} = 1/3(Z_{t-1} + Z_t + Z_{t+1})$, where $\{Z_t\} \sim \operatorname{WN}(0,\sigma^2)$. Show that $\{Y_t\}$ is a stationary process.
It's easy to prove the expectation isn't depending on time as $\{Z_t\} \sim \operatorname{WN} (0,\sigma^2)$ but how to prove that $\gamma_X(t+h,t)$ is independent of $t$ for each $h$?
That the autocovariance function is time-independent means that $$ \gamma_X(t+h,t) = \gamma_X(s+h,s), \quad \forall t,s,h\in \mathbb Z. $$ Use the expression of the autocovariance function $\gamma_X(t,s) = E[X_t\, X_s]$ (you have just shown that the process has zero expectation independent of time) and insert the expressions for the $Y_t$'s. Since $Y_t$ is an $\operatorname{MA}(3)$ process, we will write all cases of $h$ and show that the autocovariance is independent of $t$. \begin{align} \gamma_X(t+h,t) &= E[Y_{t+h}Y_{t}] \\&= 1/3^2 E\left[(Z_{t-1+h} + Z_{t+h} + Z_{t+1+h})(Z_{t-1} + Z_{t} + Z_{t+1})\right] \\\\ &= 1/9 \begin{cases} E[Z_{t-1}^2] + E[Z_{t}^2] + E[Z_{t+1}^2], & h = 0\\ E[Z_{t}^2] + E[Z_{t+1}^2], & h = 1\\ E[Z_{t+1}^2],& h = 2\\ E[Z_{t-1}^2] + E[Z_{t}^2], & h = -1 \\ E[Z_{t-1}^2], & h = -2 \\ 0, & \lvert h\rvert >2. \end{cases} \end{align} Now use that the process $\{Z_t\}_t$ is a white noise process with finite variance and insert these. We then see that $\gamma_X$ is independent of $t$.