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Short Communications

Necessary and Sufficient Conditions for Asymptotic Stability of Linear Stochastic Systems

R. Z. Khas'minskii

Moscow

Abstract: 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).

 English version: Theory of Probability and its Applications, 1967, 12:1, 144–147

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