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 Fundam. Prikl. Mat.: Year: Volume: Issue: Page: Find

 Fundam. Prikl. Mat., 1995, Volume 1, Issue 4, Pages 1009–1018 (Mi fpm119)

On asymptotic behavior of some class of random matrix iterations

A. Yu. Plakhov

Institute for Physico-Technical Problems

Abstract: In the paper iterations $J_{m+1}=J_m-\varepsilon J_mL_{S_m}J_m$, $m=0,1,2,\ldots$; $\varepsilon>0$ are considered. $J_m$ and $L_{S_m}$ are selfadjoint operators on $\mathbb R^N$, $L_{S_m}=(\cdot,S_m)S_m$, with $S_m$ being independent identically distributed random vectors which satisfy some additional conditions. Initial opetator $J_0$ is nonrandom. Asymptotic behavior of the rescaled operator $\tilde{J_m}=\|J_m\|^{-1}J_m$ is examined. Problems of this type appear in neural network theory when studying REM sleep phenomenon. It is proven that one of the following three relations holds almost surely: I. $\lim_{m\to\infty}\tilde{J}_m=P_{\mathcal L}$; II. $\lim_{m\to\infty}\tilde{J}_m=-P_{\xi}$; III. $J_m=0$ starting from some $m_0$; here $P_{\mathcal L}$ and $P_{\xi}$ are orthogonal projectors on random subspace $\mathcal L\subset\mathbb R^N$ and one-dimensional subspace spanned by random nonzero vector $\xi$, respectively. Denote $P_+(\varepsilon)$ and $P_-(\varepsilon)$ the probabilities of asymptotic behaviors I and II. For $J_0$ being nonzero positive semidefinite it is shown that $\lim_{\varepsilon\to+0}P_+(\varepsilon)=1$, $\lim_{\varepsilon\to+\infty}P_-(\varepsilon)=1$, but if $J_0$ has at least one negative eigenvalue, then $P_-(\varepsilon)\equiv1$.

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UDC: 519.21.219.5

Citation: A. Yu. Plakhov, “On asymptotic behavior of some class of random matrix iterations”, Fundam. Prikl. Mat., 1:4 (1995), 1009–1018

Citation in format AMSBIB
\Bibitem{Pla95} \by A.~Yu.~Plakhov \paper On asymptotic behavior of some class of random matrix iterations \jour Fundam. Prikl. Mat. \yr 1995 \vol 1 \issue 4 \pages 1009--1018 \mathnet{http://mi.mathnet.ru/fpm119} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1791625} \zmath{https://zbmath.org/?q=an:0871.60006}