Nonnegative chainable matrices and Kolmogorov's condition (Q2146529)

Consider a non-negative matrix \(A=(a_{ik})\). Two positive entries \(a_{ik}\) and \(a_{pq}\) are said to form a link if either \(i=p\) or \(k=q\), i.e., the entries are in either the same row or the same column. The matrix \(A\) is said to be chainable if any two positive entries can be connected by a sequence of linked positive elements, and it is said to be indecomposobale if there is no permutation matrix \(P\) such that \(PAP^{-1}\) is block triangular. The main result of the present paper is that a non-negative matrix is indecomposable and chainable if and only if it satisfies Kolmogorov's condition, under which the Markov chain determined by a stochastic matrix satisfies the multidimensional local limit theorem. The authors also study further properties of indecomposable chainable matrices, and examples of such matrices.