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Edit: Since the geometric approach did not work, I try now another approach: phrasing the problem as a quadratic programme. Probabilistic version. Let $x=(x_1,x_2, \ldots) $ be an ergodic random ...

Let $A_1$ be the $N \times N$-matrix for which $a_{i,j} = 1$ for $i=j$ and 0 otherwise. Let $A_2$ be the matrix for which $a_{i,j}=1$ for $|i-j| \leq 1$ and 0 otherwise. Similarly define $A_3$ (which ...