The definition for a strict stationary process is that the finite-dimensional distributions are invariant over time translations:
$F(X_{1},..,X{n}; t_{1},...,t_{n})=F(X_{1},..,X{n}; t_{1+\tau},...,t_{n+\tau}) \quad \forall \tau,n$
My textbook asks me to prove that if $Z$ is strictly stationary then it's mean and variance are thus constants using $X_{1},...,X_{m}$ cross-moment , but i can't manage to do it. I'd like some help please.
The definition of stationarity implies that all the random variables $X_t$ are identically distributed. Therefore they have the same (if they exist) mean and variance.