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On the convergence of moments in the almost sure central limit theorem for stochastic approximation algorithms

Published online by Cambridge University Press:  08 February 2013

Peggy Cénac
Peggy Cénac*
Affiliation:
Institut de Mathématiques de Bourgogne, IMB UMR 5584 CNRS, 9 rue Alain Savary, BP 47870, 21078 Dijon Cedex, France. peggy.cenac@u-bourgogne.fr
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Abstract

We study the almost sure asymptotic behaviour of stochastic approximation algorithms for the search of zero of a real function. The quadratic strong law of large numbers is extended to the powers greater than one. In other words, the convergence of moments in the almost sure central limit theorem (ASCLT) is established. As a by-product of this convergence, one gets another proof of ASCLT for stochastic approximation algorithms. The convergence result is applied to several examples as estimation of quantiles and recursive estimation of the mean.

Keywords

Stochastic approximation algorithmsalmost sure central limit theoremmartingale transformsmoments

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Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 179 - 194
Copyright
© EDP Sciences, SMAI, 2013

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