  We consider Markov chains with random transition probabilities which,
moreover, fluctuate randomly with time. We describe such a system by a product
of stochastic matrices, $U(t)=M_t\cdots M_1$, with the factors $M_i$ drawn
independently from an ensemble of random Markov matrices, whose columns are
independent Dirichlet random variables. The statistical properties of the
columns of $U(t)$, its largest eigenvalue and its spectrum are obtained exactly
for $N=2$ and numerically investigated for general $N$. For large $t$, the
columns are Dirichlet-distributed, however the distribution is different from
the initial one. As for the spectrum, we find that the eigenvalues converge to
zero exponentially fast and investigate the statistics of the largest Lyapunov
exponent, which is well approximated the a Gamma distribution. We also observe
a concentration of the spectrum on the real line for large $t$.
