Geodesic
Necessary and Sufficient Conditions for Asymptotic Stability of Linear Stochastic Systems
Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 1, pp. 167-172

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In this paper we consider the following two problems. (1) Let $x_0$ be an arbitrary element of $E_l$, $x_n=A_nx_{n-1}$ and $A_1,A_2\dots$ be a sequence of equidistributed independent random matrices. When $\mathbf P\{|x_n|\to0\}=1$? (2) What are the conditions for the solutions of equation (3) to tend to zero with probability 1 as $t\to\infty$? The answers to these questions are given in terms of the invariant measure of some auxiliary Markov process. In the case of problem (2) and $l=2$ the density of this measure is given by (10). 
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     author = {R. Z. Khas'minskii},
     title = {Necessary and {Sufficient} {Conditions} for {Asymptotic} {Stability} of {Linear} {Stochastic} {Systems}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {167--172},
     publisher = {mathdoc},
     volume = {12},
     number = {1},
     year = {1967},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1967_12_1_a18/}
}
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R. Z. Khas'minskii. Necessary and Sufficient Conditions for Asymptotic Stability of Linear Stochastic Systems. Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 1, pp. 167-172. http://geodesic.mathdoc.fr/item/TVP_1967_12_1_a18/