  We consider the ergodicity and consensus problem for a discrete-time linear
dynamic model driven by random stochastic matrices, which is equivalent to
studying these concepts for the product of such matrices. Our focus is on the
model where the random matrices have independent but time-variant distribution.
We introduce a new phenomenon, the infinite flow, and we study its fundamental
properties and relations with the ergodicity and consensus. The central result
is the infinite flow theorem establishing the equivalence between the infinite
flow and the ergodicity for a class of independent random models, where the
matrices in the model have a common steady state in expectation and a feedback
property. For such models, this result demonstrates that the expected infinite
flow is both necessary and sufficient for the ergodicity. The result is
providing a deterministic characterization of the ergodicity, which can be used
for studying the consensus and average consensus over random graphs.
