Let $t=0,1,2,3,...$ a discrete time index. Consider the following process
where $\gamma(t_1,t_2)$ indicates the autocovariance function. I want to prove that the process (although being autocovariance stationary) is not strongly stationary, i.e. the distribution of the variables $Z_t$ depends on time. Any help?
In general, if two random variables have all their moments to be the same, it does not follow that they have the same distributions. See here for some counterexamples.
You could however check whether the charactristic function of $Z_t$ is independent of $t$ in your case. For $Z_t$ we have $$\phi_{Z_t}(\theta)=\int_0^{2\pi}\frac{\cos tx}{2\pi}\exp(i\theta x)dx=\frac{i\theta \left(1-e^{2i\pi\theta}\right)}{2\pi\left(\theta^2-t^2\right)},$$ which is a function of $t$. Therefore the distribution of $Z_t$ depends on $t$. You could also look the distributions of $Z_1$ and $Z_2$ which are rather simple, but tedious to derive, and see that they are different.