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Teoriya Veroyatnostei i ee Primeneniya, 1967, Volume 12, Issue 1, Pages 167–172 (Mi tvp698)  

This article is cited in 258 scientific papers (total in 258 papers)

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, Volume 12, Issue 1, Pages 144–147
DOI: https://doi.org/10.1137/1112019
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: R. Z. Khas'minskii, “Necessary and Sufficient Conditions for Asymptotic Stability of Linear Stochastic Systems”, Teor. Veroyatnost. i Primenen., 12:1 (1967), 167–172; Theory Probab. Appl., 12:1 (1967), 144–147
Citation in format AMSBIB
\Bibitem{Kha67}
\by R.~Z.~Khas'minskii
\paper Necessary and Sufficient Conditions for Asymptotic Stability of Linear Stochastic Systems
\jour Teor. Veroyatnost. i Primenen.
\yr 1967
\vol 12
\issue 1
\pages 167--172
\mathnet{http://mi.mathnet.ru/tvp698}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=211465}
\zmath{https://zbmath.org/?q=an:0183.19403|0153.19702}
\transl
\jour Theory Probab. Appl.
\yr 1967
\vol 12
\issue 1
\pages 144--147
\crossref{https://doi.org/10.1137/1112019}
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  • This publication is cited in the following 258 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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