  The paper deals with the convergence properties of the products of random
(row-)stochastic matrices. The limiting behavior of such products is studied
from a dynamical system point of view. In particular, by appropriately defining
a dynamic associated with a given sequence of random (row-)stochastic matrices,
we prove that the dynamics admits a class of time-varying Lyapunov functions,
including a quadratic one. Then, we discuss a special class of stochastic
matrices, a class $\Pstar$, which plays a central role in this work. We then
introduce balanced chains and using some geometric properties of these chains,
we characterize the stability of a subclass of balanced chains. As a special
consequence of this stability result, we obtain an extension of a central
result in the non-negative matrix theory stating that, for any aperiodic and
irreducible row-stochastic matrix $A$, the limit $\lim_{k\to\infty}A^k$ exists
and it is a rank one stochastic matrix. We show that a generalization of this
result holds not only for sequences of stochastic matrices but also for
independent random sequences of such matrices.
