We consider the ergodicity and consensus problem for a discrete-time linear
dynamic model driven by random matrices, which is equivalent to studying these
concepts for the product of random matrices. Our focus is on the model where
the matrices are "stochastic". We introduce a new phenomena, the infinite flow,
and we study its fundamental properties and relations with the ergodicity and
consensus. We establish several new and important results. The central result
of this work is the infinite flow theorem establishing the role of infinite
flow in the ergodicity of a general independent random model. Through the use
of infinite flow, we show that the ergodicity of the model is equivalent to the
ergodicity of the expected model when the matrices in the model have a common
steady state in expectation and a feedback property. This result demonstrates
that for such models, the expected infinite flow is both necessary and
sufficient for the ergodicity. The result is providing us with a powerful
deterministic characterization of the ergodicity, which renders a new elegant
tool that can be used for studying the consensus and average consensus over
random graphs, as well as random consensus algorithms.