(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

## On asymptotic behavior of some class of random matrix iterations

A.Yu.Plakhov

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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