The convergence of general products of matrices and the weak ergodicity of Markov chains (Q1301309)

The paper concerns the infinite general products of real or complex matrices. General means that the matrices to form the product are taken in arbitrary order from an infinite sequence of matrices \(\{A_i\}\). In the case of stochastic matrices, such general products have been considered by \textit{E. Seneta} [Non-negative matrices and Markov chains, Springer (1981; Zbl 0471.60001)]. In two main theorems, the authors give sufficient conditions for such a product to be bounded and to be convergent to zero, respectively. These conditions use the matrix submultiplicative norm \(\mu\) and are based on the convergence of \(\sum_i [\max (\mu(A_i),1)- 1]\) and divergence of \(\sum_i [1-\min (\mu(A_i),1)]\), respectively. The proofs are based on classical theorems related with the products of positive numbers. The results obtained are used to deduce a condition for weak ergodicity of inhomogeneous Markov chain. Also the ergodicity coefficient based on the matrix norm is compared with other ergodicity coefficients considered, for example, in a paper by \textit{A. Rhodius} [Linear Algebra Appl. 194, 71-83 (1993; Zbl 0792.15009)].