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February, 1982 Almost Sure Approximations to the Robbins-Monro and Kiefer-Wolfowitz Processes with Dependent Noise
David Ruppert
Ann. Probab. 10(1): 178-187 (February, 1982). DOI: 10.1214/aop/1176993921

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.

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David Ruppert. "Almost Sure Approximations to the Robbins-Monro and Kiefer-Wolfowitz Processes with Dependent Noise." Ann. Probab. 10 (1) 178 - 187, February, 1982. https://doi.org/10.1214/aop/1176993921

Information

Published: February, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0485.62083
MathSciNet: MR637384
Digital Object Identifier: 10.1214/aop/1176993921

Subjects:
Primary: 62L20
Secondary: 60F15

Keywords: almost sure invariance principle , Dependent random variables , Kiefer-Wolfowitz process , Robbins-Monro process , stochastic approximation

Rights: Copyright © 1982 Institute of Mathematical Statistics

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Vol.10 • No. 1 • February, 1982
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