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Stochastic matrix-valued cocycles and non-homogeneous Markov chains
We prove weak ergodicity theorems for non - homogeneous Markov
chains $\{X_\nu\}_{\nu\geq 0}$ taking values in a finite state space
$S=\{1,\cdots,n\}$ for which the family of transition matrices
$\{g(x)\}_{x\in X}$ is generated from some underlying topological or
measurable dynamical system $f:X\to X$. Using the projective metric
of Hilbert on $\mathcal{S}=\{(x_1,\cdots,x_n)\in\mathbb R ^n :
x_i\geq 0, x_1+\cdots+x_n=1\}$, the space of distributions, we form
the skew-product $T:X\times\mathcal{S}\to X\times\mathcal{S}$
defined by $T(x,p)=(f(x),g(x)p)$ and show that, for continuous $g$
positive on some set, weak ergodicity for such processes is a result
of the existence of a map $\gamma:X\to\mathcal{S}$ whose graph is
attracting and invariant under $T$. Some results on random
compositions of non-expansive maps are obtained on the way.