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

Koval’, V. O.

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.

Reviewer: Mikhail Moklyachuk (Kyïv)

MSC:

62L20	Stochastic approximation
60F15	Strong limit theorems

Keywords:

stochastic difference equation; rate of convergence

Citations:

Zbl 0049.36601