Suppose $X_n^{(1)}=\lambda_1 X_{n-1}^{(1)}+\mu_1+\epsilon_n^{(1)}$ and $X_{n}^{(2)}=\lambda_2X_{n-1}^{(2)}+\mu_2+\epsilon_n^{(2)}$ where $|\lambda_i|<1$ for $i=1,2$ and $\epsilon_n^{(i)}$ are standard Gaussian independent across $n$ and $i$.
What I am trying to understand is how each of the four parameters $\mu_1,\mu_2,\lambda_1,\lambda_2$ affects the statistical difference between the two processes $X_n^{(1)}$ and $X_n^{(2)}$. By looking at the Kullback-Liebler divergence it can be seen that if say $\mu_2>\mu_1>0$ then the processes are further apart for larger $\lambda_2>0$, which makes sense intuitively, as greater positive correlation makes the mean difference more contrast. However, if $\lambda_2<0$ then you'd think that the processes we get closer for smaller (below zero) $\lambda_2$, but that's not exactly true. As soon as $\lambda_2$ drops below a certain (computable) negative value, the difference between the processes stops decaying and starts increasing again. Is there any way to explain this intuitively, w/o using the Kullback-Liebler divergence formula?
Thanks.