On rate of convergence of procedures of stochastic approximation under some conditions of dependence.(Ukrainian, English)Zbl 1022.62068

Teor. Jmovirn. Mat. Stat. 65, 60-66 (2001); translation in Theory Probab. Math. Stat. 65, 69-76 (2002).
Let $$(\mathbb{B},\|\cdot\|)$$ be a separable Banach space. Let $$Y_n$$ be a solution to the nonlinear stochastic difference equation $Y_n=Y_{n-1}-a_n(F(Y_{n-1})+\Psi_n)+b_nV_n,\quad n\geq 1,\quad Y_0\in\mathbb{B},$ where $$\{a_n,n\geq 1\}$$ is a sequence of positive numbers such that $$\lim_{n\to\infty}a_n=0$$, $$\sum_{n=1}^{\infty}a_n=\infty$$; $$\{b_n,\;n\geq 1\}$$ is a sequence of real numbers; $$\{V_n,\;n\geq 1\}$$ is a sequence of random elements from $$\mathbb{B}$$; $$\{\Psi_n,n\geq 1\}$$ is a sequence of random elements from $$\mathbb{B}$$ such that $$\|\Psi_n\|\to 0,\;n\to\infty$$ a.e.; the equation $$F(x)=0$$ has a solution $$\theta$$.
The author investigates the rate of convergence of $$Y_n$$ to $$\theta$$ as $$n\to\infty$$ under some general conditions on the sequences $$\{a_n,n\geq 1\}$$, $$\{b_n,n\geq 1\}$$ and some assumptions on dependence of the random perturbations $$\{V_n,\;n\geq 1\}$$.
The Robbins-Monro and the Kiefer-Wolfowitz [see J. Kiefer and J. Wolfowitz, Ann. Math. Stat. 23, 462-466 (1952; Zbl 0049.36601)] procedures of stochastic approximation are particular cases of the proposed results.

MSC:

 62L20 Stochastic approximation 60F15 Strong limit theorems

Zbl 0049.36601