Almost Sure Approximations to the Robbins-Monro and Kiefer-Wolfowitz Processes with Dependent Noise

Abstract

We study a recursive algorithm which includes the multidimensional Robbins-Monro and Kiefer-Wolfowitz processes. The assumptions on the disturbances are weaker than the usual assumption that they be a martingale difference sequence. It is shown that the algorithm can be represented as a weighted average of the disturbances. This representation can be used to prove asymptotic results for stochastic approximation procedures. As an example, we approximate the one-dimensional Kiefer-Wolfowitz process almost surely by Brownian motion and as a byproduct obtain a law of the iterated logarithm.