1995, VOLUME 1, NUMBER 4, PAGES 1009-1018

On asymptotic behavior of some class of random matrix iterations


In the paper iterations $J_{m+1} = J_m - \varepsilon J_m L_{S_m} J_m$, m = 0,1,2,...$; $\varepsilon > 0$ are considered. Jm and LSm are selfadjoint operators on $\mathbb{R}^N$, $L_{S_m} = (\cdot, S_m) S_m$, with Sm being independent identically distributed random vectors which satisfy some additional conditions. Initial opetator J0 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. $\lim_{m\rightarrow\infty} \tilde{J}_m = P_{\mathcal{L}}$; II. $\lim_{m\rightarrow\infty} \tilde{J}_m = -P_{\xi}$; III. Jm = 0 starting from some m0; 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 J0 being nonzero positive semidefinite it is shown that $\lim_{\varepsilon\rightarrow+0} P_+(\varepsilon) = 1$, $\lim_{\varepsilon\rightarrow+\infty} P_-(\varepsilon) = 1$, but if J0 has at least one negative eigenvalue, then $P_-(\varepsilon) \equiv 1$.

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