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Damping perturbations of dynamic systems and convergence conditions for recursive stochastic procedures
A. P. Korostelev Moscow
Abstract:
Let dynamic system
$\dot x_t=b(x_t)$ in
$R^d$ has stable equilibrium state at the point 0. Random perturbations of this system are considered as
$dX_t=b(X_t)\,dt+d\zeta(t,X_t)$, where
$\zeta(t,x)$ for any
$x$ is the process with independent increments which damps when
$t\to\infty$. Following [9] we show that
$X_t$-paths leave an arbitrary domain
$D_0$ containing point 0 during time
$T$ after moment
$t_0$ with probability the main term of which for
$t_0\to\infty$ has the form
$$
\exp\{-g_T(t_0)V_T(D_0)\},\quad g_t(t_0)\to\infty,\quad V_T(D_0)>0.
$$
In many cases this probability may be estimated from above and from below by
$\exp\{-g(t_0)(V(D_0)\pm h)\}$ with arbitrary small
$h>0$. In such a case either
$X_t$-paths leave the domain
$D_0$ with probability 1 after any moment
$t_0$ or stay in
$D_0$ with probability which tends to 1 when
$t_0\to\infty$. These two possibilities depend on the divergence or convergence of the integral
$$
\int_0^{\infty}\exp\{-g(t_0)V(D_0)\}\,dt_0.
$$
The results are applied to the investigation of convergence conditions for some stochastic recursive procedures. In a number of cases for Robbins–Monro and Kiefer–Wolfowitz procedures the necessary and sufficient conditions are obtained.
Received: 18.04.1977
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