We know any stationary process ${(X_n)}_{n \in \mathbb{Z}}$ can be represented as $X_n(\omega)=X(\phi^n\omega)$. where we take the shift $\phi$ on the canonical space of the process and $X$ maps a sequence to its central coordinate.
Question: How can we show that the invariant $\sigma$-field is then the tail $\sigma$-field $\cap_{n \leq 0}\sigma(X_m, m\leq n)$. see https://math.stackexchange.com/posts/219995/edit
Comments:
When the process is one side infinite that is change $Z$ to $N$. Then it's easy to show this because when shift a invariant set you can get rid of the first coordinate. that is if $A=\phi^{-1}A, A\in \sigma(X_0,X_1,...)$, then $A\in \sigma(X_1,X_2,...)$.
But when the process is double infinite the hard part is we don't miss one coordinate when shift an event.