  We study the ergodicity of backward product of stochastic and doubly
stochastic 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 convergence result for
ergodic doubly stochastic chains. We also investigate the limiting behavior of
a doubly stochastic chain and show that the product of doubly stochastic
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.
