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/
[1] , and , Adaptive Algorithms and Stochastic Approximations. Springer-Verlag, New York, Appl. Math. 22 (1990). | MR | Zbl
[2] , On the convergence of moments in the almost sure central limit theorem for martingales with statistical applications. Stoc. Proc. Appl. 111 (2004) 157-173. | MR | Zbl
[3] and , A moment approach for the almost sure central limit theorem for martingales. Stud. Sci. Math. Hung. (2006). | Zbl
[4] , and , On the almost sure central limit theorem for vector martingales : Convergence of moments and statistical applications. J. Appl. Probab. 46 (2009) 151-169. | MR | Zbl
[5] , An almost everywhere central limit theorem. Math. Proc. Cambridge Philos. Soc. 104 (1988) 213-246. | MR | Zbl
[6] , Version forte du théorème de la limite centrale fonctionnel pour les martingales. C. R. Acad. Sci. Paris Sér. I Math. 323 (1996) 195-198. | MR | Zbl
[7] , Invariance principles with logarithmic averaging for martingales. Stud. Sci. Math. Hung. 37 (2001) 21-52. | MR | Zbl
[8] and , Théorèmes limites avec poids pour les martingales vectorielles. ESAIM : PS 4 (2000) 137-189 (electronic). | Numdam | MR | Zbl
[9] , and , Génèralisation du théorème de la limite centrale presque-sûr pour les martingales vectorielles. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998) 229-232. | MR | Zbl
[10] , Random Iterative Methods. Springer-Verlag (1997). | MR | Zbl
[11] and , Stochastic approximation and large deviations : Upper bounds and w.p.l convergence. SIAM J. Control Optim. 27 (1989) 1108-1135. | MR | Zbl
[12] , An introduction to probability theory and its applications II. John Wiley, New York (1966). | MR | Zbl
[13] and , Martingale Limit Theory and Its Application. Academic Press, New York, NY (1980). | MR | Zbl
[14] and , Exact bounds for the rate of convergence of stochastic approximation procédures. Stoc. Anal. Appl. 16 (1998) 501-515. | MR | Zbl
[15] and , Stochastic Approximation Methods for Constrained and Unconstrained Systems. Springer-Verlag, Berlin (1978). | MR | Zbl
[16] and , A note on the almost sure central limit theorem. Stat. Probab. Lett. 9 (1990) 201-205. | MR | Zbl
[17] and , Recursive computation of the invariant distribution of a diffusion. Bernoulli 8 (2002) 367-405. | MR | Zbl
[18] and , Recursive computation of the invariant distribution of a diffusion : the case of a weakly mean reverting drift. Stoch. Dyn. 3 (2003) 435-451. | MR | Zbl
[19] , About the averaging approach schemes for stochastic approximations. Math. Methods Stat. 2 (1993) 295-315. | MR | Zbl
[20] and , Averaging for estimating covariances in stochastic approximation. Math. Methods Stat. 3 (1994) 244-266. | MR | Zbl
[21] and , Some results about averaging in stochastic approximation. Metrika 42 (1995) 153-171. | MR | Zbl
[22] , Lecture notes on almost sure limit theorems. Publications IRMA 54 (2001) 1-25.
[23] , Almost sure limit theorem for martingales, in Limit theorems in probability and statistics II (Balatonlelle, 1999). János Bolyai Math. Soc., Budapest (2002) 367-390. | MR | Zbl
[24] , and , Stochastic Approximation and Optimization of Random Systems. Birkhäuser, Boston (1992). | MR | Zbl
[25] and , A companion for the Kiefer-Wolfowitz-Blum stochastic approximation algorithm. Ann. Stat. (2007). | Zbl
[26] , On the almost sure asymptotic behaviour of stochastic algorithms. Stoch. Proc. Appl. 78 (1998) 217-244. | MR | Zbl
[27] , An almost sure central limit theorem for stochastic approximation algorithms. J. Multivar. Anal. 71 (1999) 76-93. | MR | Zbl
[28] and , A stochastic approximation method. Ann. Math. Stat. 22 (1951) 400-407. | MR | Zbl
[29] , On strong versions of central limit theorem. Math. Nachr. 137 (1988) 249-256. | MR | Zbl
[30] , Asymptotic normality for a vector stochastic difference equation with applications in stochastic approximation. J. Multivar. Anal. 57 (1996) 101-118. | MR | Zbl
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