Let $Y_1:=(X_i)_{i \in \mathbb Z}$ be a family of random variables that are identically distributed but not necessarily independent.

We can then also define the shifted sequence $Y_2:=(X_{i+1})_{i \in \mathbb Z}.$

If the $X_i$ were also independent then $f(Y_1)$ would have the same law as $f(Y_2).$

In particular, we can conclude from this that also $$\mathbb E(f(Y_1))=\mathbb E(f(Y_2)).$$

I am wondering whether we can still conclude $$\mathbb E(f(Y_1))=\mathbb E(f(Y_2))$$

if we assume the $X_i$ only to be identically distributed but not necessarily independent.