Consider the lattice $\mathbb Z^2$ and take iid random variables $Y_e$ on all edges $e$ of the graph. 

We then define random variables $X_i:=\sum_{e \text{ adjacent to } i}Y_e.$ 

In other words: For every vertex we define a random variable $X_i$ that is the sum of the $Y_e$ on its neighbouring edges.

I ask: Is there a probability space $(\Omega,\mathcal A, \mathbb P)$ such that this stochastic process $(X_i)$ is ergodic with respect to the standard shift $(T_i)_{i \in \mathbb Z}$ where $(T_i f)(z)=f(z-i)$? 
If the $X_i$ were independent this would be easy as the product measure would do it. But they are only almost independent in the sense that $X_i$ and $X_j$ are independent if $\vert i-j\vert \ge 2$.


I recall for you: 

A stochastic process is called ergodic if there is an ergodic family of measure preserving transformations $(T_i)$ such that $X_i(T_j\omega)= X_{i-j}(\omega).$

Now, the shift is a reasonable candidate to be measure preserving and it satisfies also the property $X_i(T_j\omega)= X_{i-j}(\omega)$

Can we also realize it as an ergodic family?