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.
Mots clés : stochastic approximation algorithms, almost sure central limit theorem, martingale transforms, moments
@article{PS_2013__17__179_0,
author = {C\'enac, Peggy},
title = {On the convergence of moments in the almost sure central limit theorem for stochastic approximation algorithms},
journal = {ESAIM: Probability and Statistics},
pages = {179--194},
publisher = {EDP-Sciences},
volume = {17},
year = {2013},
doi = {10.1051/ps/2011155},
mrnumber = {3021314},
zbl = {1290.60019},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2011155/}
}
TY - JOUR AU - Cénac, Peggy TI - On the convergence of moments in the almost sure central limit theorem for stochastic approximation algorithms JO - ESAIM: Probability and Statistics PY - 2013 SP - 179 EP - 194 VL - 17 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2011155/ DO - 10.1051/ps/2011155 LA - en ID - PS_2013__17__179_0 ER -
%0 Journal Article %A Cénac, Peggy %T On the convergence of moments in the almost sure central limit theorem for stochastic approximation algorithms %J ESAIM: Probability and Statistics %D 2013 %P 179-194 %V 17 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2011155/ %R 10.1051/ps/2011155 %G en %F PS_2013__17__179_0
Cénac, Peggy. On the convergence of moments in the almost sure central limit theorem for stochastic approximation algorithms. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 179-194. doi : 10.1051/ps/2011155. http://www.numdam.org/articles/10.1051/ps/2011155/
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