Almost sure convergence and asymptotical normality of a generalization of Kesten's stochastic approximation algorithm for multidimensional case

TL;DR

This paper proves the almost sure convergence and asymptotic normality of a multidimensional generalization of Kesten's stochastic approximation algorithm, which adaptively adjusts steps based on the sign of scalar products of estimate increments.

Contribution

It introduces a novel step adjustment rule for multidimensional stochastic approximation, enhancing convergence properties and behavior acceleration.

Findings

Proves almost sure convergence of the proposed algorithm.

Establishes asymptotic normality of the estimates.

Demonstrates accelerated entrance into stochastic behavior.

Abstract

It is shown the almost sure convergence and asymptotical normality of a generalization of Kesten's stochastic approximation algorithm for multidimensional case. In this generalization, the step increases or decreases if the scalar product of two subsequente increments of the estimates is positive or negative. This rule is intended to accelerate the entrance in the `stochastic behaviour' when initial conditions cause the algorithm to behave in a `deterministic fashion' for the starting iterations.