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

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References

A. Benveniste, M. Métivier and P. Priouret, Adaptive Algorithms and Stochastic Approximations. Springer-Verlag, New York, Appl. Math. 22 (1990).
Bercu, B., On the convergence of moments in the almost sure central limit theorem for martingales with statistical applications. Stoc. Proc. Appl. 111 (2004) 157–173. Google Scholar
B. Bercu and J.-C. Fort, A moment approach for the almost sure central limit theorem for martingales. Stud. Sci. Math. Hung. (2006).
Bercu, B., Cènac, P. and Fayolle, G., On the almost sure central limit theorem for vector martingales : Convergence of moments and statistical applications. J. Appl. Probab. 46 (2009) 151–169. Google Scholar
Brosamler, G.A., An almost everywhere central limit theorem. Math. Proc. Cambridge Philos. Soc. 104 (1988) 213–246. Google Scholar
Chaâbane, F., Version forte du théorème de la limite centrale fonctionnel pour les martingales. C. R. Acad. Sci. Paris Sér. I Math. 323 (1996) 195–198. Google Scholar
Chaâbane, F., Invariance principles with logarithmic averaging for martingales. Stud. Sci. Math. Hung. 37 (2001) 21–52. Google Scholar
Chaâbane, F. and Maâouia, F., Théorèmes limites avec poids pour les martingales vectorielles. ESAIM : PS 4 (2000) 137–189 (electronic). Google Scholar
Chaâbane, F., Maâouia, F. and Touati, A., 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. Google Scholar
M. Duflo, Random Iterative Methods. Springer-Verlag (1997).
Dupuis, P. and Kushner, H.J., Stochastic approximation and large deviations : Upper bounds and w.p.l convergence. SIAM J. Control Optim. 27 (1989) 1108–1135. Google Scholar
W. Feller, An introduction to probability theory and its applications II. John Wiley, New York (1966).
P. Hall and C.C. Heyde, Martingale Limit Theory and Its Application. Academic Press, New York, NY (1980).
Koval, V. and Schwabe, R., Exact bounds for the rate of convergence of stochastic approximation procédures. Stoc. Anal. Appl. 16 (1998) 501–515. Google Scholar
H.J. Kushner and D.S. Clark, Stochastic Approximation Methods for Constrained and Unconstrained Systems. Springer-Verlag, Berlin (1978).
Lacey, M. and Phillip, W., A note on the almost sure central limit theorem. Stat. Probab. Lett. 9 (1990) 201–205. Google Scholar
Lamberton, D. and Pagès, G., Recursive computation of the invariant distribution of a diffusion. Bernoulli 8 (2002) 367–405. Google Scholar
Lamberton, D. and Pagès, G., Recursive computation of the invariant distribution of a diffusion : the case of a weakly mean reverting drift. Stoch. Dyn. 3 (2003) 435–451. Google Scholar
Le Breton, A., About the averaging approach schemes for stochastic approximations. Math. Methods Stat. 2 (1993) 295–315. Google Scholar
Le Breton, A. and Novikov, A., Averaging for estimating covariances in stochastic approximation. Math. Methods Stat. 3 (1994) 244–266. Google Scholar
Le Breton, A. and Novikov, A., Some results about averaging in stochastic approximation. Metrika 42 (1995) 153–171. Google Scholar
Lifshits, M.A., Lecture notes on almost sure limit theorems. Publications IRMA 54 (2001) 1–25. Google Scholar
M.A. Lifshits, Almost sure limit theorem for martingales, in Limit theorems in probability and statistics II (Balatonlelle, 1999). János Bolyai Math. Soc., Budapest (2002) 367–390.
L. Ljung, G. Pflug and H. Walk, Stochastic Approximation and Optimization of Random Systems. Birkhäuser, Boston (1992).
A. Mokkadem and M. Pelletier, A companion for the Kiefer–Wolfowitz–Blum stochastic approximation algorithm. Ann. Stat. (2007).
Pelletier, M., On the almost sure asymptotic behaviour of stochastic algorithms. Stoch. Proc. Appl. 78 (1998) 217–244. Google Scholar
Pelletier, M., An almost sure central limit theorem for stochastic approximation algorithms. J. Multivar. Anal. 71 (1999) 76–93. Google Scholar
Robbins, H. and Monro, S., A stochastic approximation method. Ann. Math. Stat. 22 (1951) 400–407. Google Scholar
Schatte, P., On strong versions of central limit theorem. Math. Nachr. 137 (1988) 249–256. Google Scholar
Zhu, Y., Asymptotic normality for a vector stochastic difference equation with applications in stochastic approximation. J. Multivar. Anal. 57 (1996) 101–118. Google Scholar