Geodesic
On Measures with Supports Generated by the Lie Algebra
Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 1, pp. 154-160

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Consider product $g(n)=g_1\dots g_n$ of $n$ independent random unimodular matrices with distribution $\mu$ (which is supposed to be absolutely continuous with respect to the Haar measure on corresponding group $G$). If these matrices are real it is possible that the distributions of $g(n)$ and $g(n+1)$ be quite different even for large $n$. This fact depends on the existence of periodicity in a Markov chain. In this paper it is proved that the periodicity cannot exist if $\mu(\exp L)>0$ where $L$ is the Lie algebra of $G$. 
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     author = {V. N. Tutubalin},
     title = {On {Measures} with {Supports} {Generated} by the {Lie} {Algebra}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {154--160},
     publisher = {mathdoc},
     volume = {12},
     number = {1},
     year = {1967},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1967_12_1_a16/}
}
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V. N. Tutubalin. On Measures with Supports Generated by the Lie Algebra. Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 1, pp. 154-160. http://geodesic.mathdoc.fr/item/TVP_1967_12_1_a16/