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Let $\Psi_n$ be a product of $n$ independent, identically distributed randommatrices $M$, with the properties that $\Psi_n$ is bounded in $n$, and that $M$has a deterministic (constant) invariant vector. Assuming that the probabilityof $M$ having only the simple eigenvalue 1 on the unit circle does not vanish,we show that $\Psi_n$ is the sum of a fluctuating and a decaying process. Thelatter converges to zero almost surely, exponentially fast as $n\to\infty$. Thefluctuating part converges in Cesaro mean to a limit that is characterizedexplicitly by the deterministic invariant vector and the spectral data of${\mathbb E}[M]$ associated to 1. No additional assumptions are made on thematrices $M$; they may have complex entries and not be invertible.  We apply our general results to two classes of dynamical systems:inhomogeneous Markov chains with random transition matrices (stochasticmatrices), and random repeated interaction quantum systems. In both cases, weprove ergodic theorems for the dynamics, and we obtain the form of the limitstates.

Infinite products of random matrices and repeated interaction dynamics