We are given \begin{align} Y_j &=a_0Z_j+a_1Z_{j-1}-a_0Z_{j-2}\\ Y_{j-1} &=a_0Z_{j-1}+a_1Z_{j-2}-a_0Z_{j-3}. \end{align}
The task to show that $Y_j$ and $Y_{j-1}$ are not independent, where $\mathbb{P}(Z_i=1)=\mathbb{P}(Z_i=-1)=\frac{1}{2}$, $\forall i\in\mathbb{Z}$.
While being obvious I found it difficult to prove it. I have tried to assume that all coefficients are equal to 1, consider the maximum of $Y_{j-1}$ and condition probability $\mathbb{P}(Y_j=y_j|Y_{j-1})$ but got stuck in derivation.
Any ideas?