This thesis mainly deals with infinite products of deterministic and random stochastic matrices and generalization to random chains and general state spaces. These products represent one of the analytical tools that is often used in various problems including distributed computation, distributed optimization and other distributed systems. In many of these problems, a common goal is to determine how a set of agents behave while there is no central coordination among them. A way for solving those problems is to reach a form of agreement by performing distributed averaging among these agents, which leads to another method for the study of products of stochastic matrices, i. e., to the study of so called weighted averaging dynamics.
In the thesis, when discussing the framework for studying a product of random stochastic matrices the concept of an infinite flow property is introduced. It is then shown that this property is necessary for ergodicity of any chain. It follows that the limiting behavior of a product of stochastic matrices is closely related to the connectivity of the infinite flow graph associated with such a chain.
It is also introduced the concept of infinite flow stability for a random chain and with it connected the feedback property. Different feedback properties are defined and their relations with each other are investigated. Using the quadratic comparison function it is shown that any chain in a certain class of adapted random chains with weak feedback property is infinite flow stable. The notion of infinite flow property is extended to absolute infinite flow property. It is proved that this stronger property is also necessary for ergodicity of any stochastic chain. It is done through the introduction of a rotational transformation of a stochastic chain with respect to a permutation chain. It is then shown that the limiting behavior of a stochastic chain is invariant under rotational transformation. Using this result it is proved that ergodicity of any doubly stochastic chain is equivalent to absolute infinite flow property. Using the rotational transformation it is also shown that any product of doubly stochastic matrices is essentially convergent.
Finally, the notion of averaging and product of stochastic matrices is extended to general measurable state spaces. In this case, the several modes of ergodicity are defined and the notion of infinite flow property is extended. It is proved, that in general state spaces this property is necessary for the weakest form of ergodicity.