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Teor. Veroyatnost. i Primenen., 1967, Volume 12, Issue 1, Pages 167–172 (Mi tvp698)  

This article is cited in 2 scientific papers (total in 2 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 $A_1,A_2…$ 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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English version:
Theory of Probability and its Applications, 1967, 12:1, 144–147

Bibliographic databases:


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://www.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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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. R. Z. Khasminskii, “Stability of regime-switching stochastic differential equations”, Problems Inform. Transmission, 48:3 (2012), 259–270  mathnet  crossref  isi
    2. L. I. Rodina, I. I. Tyuteev, “Ob asimptoticheskikh svoistvakh reshenii raznostnykh uravnenii so sluchainymi parametrami”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 26:1 (2016), 79–86  mathnet  crossref  mathscinet  elib
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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