Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, let $(E,\mathcal E$) a measurable space. Let $(X_n)_n$ be a succession of random variable from $\Omega$ to $E$. We consider the whole $(X_n)_n$ as a block function from $\Omega$ to $E^{\mathbb N^*}$, the space of all successions of elements of $E$. Let $\mathcal E^{\mathbb N^*}$ be its associated $\sigma$-algebra. Finally, define the left-shift function $\tau$.
Then $(X_n)_n: \Omega \to E^{\mathbb N^*}$ is stationary if, for every $B\in\mathcal E^{\mathbb N^*}$,
$ \mathbb P((X_n)_n^{-1}(B)) = \mathbb P((X_n)_n^{-1}(\tau(B))) $
I have the feeling that this is close to a correct definition, even if I left out a lot of details.
EDIT: Alternatively, how can I define a stationary sequence of random variables using only image-measures? I am a bit confused.