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.
Publié le : 1982-02-14
Classification:
Stochastic approximation,
Robbins-Monro process,
Kiefer-Wolfowitz process,
dependent random variables,
almost sure invariance principle,
62L20,
60F15
@article{1176993921,
author = {Ruppert, David},
title = {Almost Sure Approximations to the Robbins-Monro and Kiefer-Wolfowitz Processes with Dependent Noise},
journal = {Ann. Probab.},
volume = {10},
number = {4},
year = {1982},
pages = { 178-187},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993921}
}
Ruppert, David. Almost Sure Approximations to the Robbins-Monro and Kiefer-Wolfowitz Processes with Dependent Noise. Ann. Probab., Tome 10 (1982) no. 4, pp. 178-187. http://gdmltest.u-ga.fr/item/1176993921/