Consider a stationary and ergodic E-valued sequence $\mathbf X= (\dots, X_{-k}, \dots, X_{-1}, X_0, X_1, \dots, X_k, \dots )$ and $f: E^{\mathbb Z^+} \to \tilde E $ a measurable function. I would like to conclude that the sequence $\mathbf Y= (\dots, Y_{-k}, \dots, Y_{-1}, Y_0, Y_1, \dots, Y_k, \dots )$ defined as $Y_k =f(\dots, X_{-k}, \dots, X_{-1}, X_0, X_1, \dots, X_k)$ is also stationary and ergodic.
In Ulrich Krengel's book "Ergodic Theorems," I have found Corollary 4.2 which, to my understanding, proves that if $\mathbf X$ is stationary also $\mathbf Y$ is so.
But the corresponding ergodic part is only stated for unilateral sequences:
Note that ergodicity itself is defined before Proposition 4.3 only for unilateral sequences.
Due to my lack of familiarity with these concepts, I am not really confident about the fact that the same result holds also for bilateral sequences. Any help is welcome. Thanks in advance.