Non-singularity conjecture for the partial-sums coefficient matrix

Let $S_N$ be the sum of the discrete random variables considered in the system, and let $\mathbb{E}S_N$ denote its expectation. The coefficient matrix of is the matrix determining the unknown initial probabilities. Non-singularity conjecture. The coefficient matrix in is non-singular whenever $\mathbb{E}S_N<0$. Numerical experiments support this claim, while a general proof remains elusive when the terms $b_k(x)$ in the recurrence are not identically zero.