A time series $X_t$ is called $m$-dependent if the random vectors $(\ldots , X_{t-1}, X_t)$ and $(X_{t+m+1}, X_{t+m+2}, \ldots )$ are independent for every $t \in \mathbb{Z}$.
Despite it is intuitively logical that the moving average $X_t = Z_t + \theta Z_{t-1}$, with $Z_t$ an i.i.d. sequence, is 1-dependent, I am having trouble to provide the proof of this fact.
An approach could be to take a look at the sigma-algebras generated by the two random vector $(\ldots , X_{t-1}, X_t)$ and $(X_{t+2}, X_{t+3}, \ldots )$. However I do not see how to proceed?
Can anybody give a reasonable approach to prove the above statement?