Geodesic
On the products of random matrices and operators
Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 2, pp. 361-370

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Let $\xi_1,\xi_2,\dots$ be a stationary ergodic Markovian process on a measurable space $\Xi$ and $X$ be a measurable mapping of $\Xi$ into the group $SL(m,R)$. We prove that, under some conditions, the norm of the product $$ X(\xi_1)X(\xi_2)\dots X(\xi_n) $$ of random unimodular matrices grows exponentially with probability 1 (Theorem 1). The proof is based on some facts from the theory of unitary representations of the group $SL(m,R)$ and on the theorem on the exponential decrease of the mean of the product of random unitary operators on a separable Hilbert space (Theorem 2). 
@article{TVP_1979_24_2_a8,
     author = {A. D. Vircer},
     title = {On the products of random matrices and operators},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {361--370},
     publisher = {mathdoc},
     volume = {24},
     number = {2},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1979_24_2_a8/}
}
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A. D. Vircer. On the products of random matrices and operators. Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 2, pp. 361-370. http://geodesic.mathdoc.fr/item/TVP_1979_24_2_a8/