FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(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
$P_-(\varepsilon) \equiv 1$.
All articles are
published in Russian.
Location: http://mech.math.msu.su/~fpm/eng/95/954/95411.htm
Last modified: October 15, 1997.