On backward product of stochastic matrices

We study the ergodicity of backward product of stochastic and doubly stochas-tic matrices by introducing the concept of absolute infinite flow property. We show that this property is necessary for ergodicity of any chain of stochastic matrices, by defining and exploring the properties of a rotational transformation for a stochastic chain. Then, we establish that the absolute infinite flow property is equivalent to ergodicity for doubly stochastic chains. Furthermore, we develop a rate of conver-gence result for ergodic doubly stochastic chains. We also investigate the limiting behavior of a doubly stochastic chain and show that the product of doubly stochas-tic matrices is convergent up to a permutation sequence. Finally, we apply the results to provide a necessary and sufficient condition for the absolute asymptotic stability of a discrete linear inclusion driven by doubly stochastic matrices.