Alternative characterization of ergodicity for doubly stochastic chains

In this paper we discuss the ergodicity of stochastic and doubly stochastic chains. We define absolute infinite flow property and show that this property is necessary for ergodicity of any stochastic chain. The proof is constructive and makes use of a rotational transformation, which we introduce and study. We then focus on doubly stochastic chains for which we prove that the absolute infinite flow property and ergodicity are equivalent. The proof of this result makes use of a special decomposition of a doubly stochastic matrix, as given by Birkhoff-von Neumann theorem. Finally, we show that a backward product of doubly stochastic matrices is convergent up to a permutation sequence and, as a result, the set of accumulation points of such a product is finite.