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Asymptotic normality of randomly truncated stochasticalgorithms

Published online by Cambridge University Press:  08 February 2013

Jérôme Lelong
Jérôme Lelong*
Affiliation:
Laboratoire Jean Kuntzmann, Université de Grenoble et CNRS, BP 53, 38041 Grenoble Cedex 9, France. [email protected]
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Abstract

We study the convergence rate of randomly truncated stochastic algorithms, which consistin the truncation of the standard Robbins–Monro procedure on an increasing sequence ofcompact sets. Such a truncation is often required in practice to ensure convergence whenstandard algorithms fail because the expected-value function grows too fast. In this work,we give a self contained proof of a central limit theorem for this algorithm under localassumptions on the expected-value function, which are fairly easy to check in practice.

Keywords

Stochastic approximationcentral limit theoremrandomly truncated stochastic algorithmsmartingale arrays.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 105 - 119
Copyright
© EDP Sciences, SMAI, 2013

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References

Arouna, B., Adaptative Monte Carlo method, a variance reduction technique. Monte Carlo Methods Appl. 10 (2004) 1–24. Google Scholar
A. Benveniste, M. Métivier and P. Priouret, Adaptive algorithms and stochastic approximations. Springer-Verlag, Berlin. Appl. Math. 22 (1990). Translated from the French by Stephen S. Wilson.
C. Bouton, Approximation Gaussienne d’algorithmes stochastiques à dynamique Markovienne. Ph.D. Thesis, Université Pierre et Marie Curie, Paris 6 (1985).
Buche, R. and Kushner, H.J., Rate of convergence for constrained stochastic approximation algorithms. SIAM J. Control Optim. 40 (2001) 1011–1041 (electronic). Google Scholar
H.-F. Chen, Stochastic approximation and its applications, Kluwer Academic Publishers, Dordrecht. Nonconvex Optim. Appl. 64 (2002).
H. Chen and Y. Zhu, Stochastic Approximation Procedure with randomly varying truncations. Scientia Sinica Series (1986).
Delyon, B., General results on the convergence of stochastic algorithms. IEEE Trans. Automat. Contr. 41 (1996) 1245–1255. Google Scholar
M. Duflo, Algorithmes stochastiques (Mathématiques et Applications). Springer (1996).
M. Duflo, Random Iterative Models. Springer-Verlag Berlin and New York (1997).
H.J. Kushner and G.G. Yin, Stochastic approximation and recursive algorithms and applications, Applications of Mathematics. Springer-Verlag, New York, 2nd edition 2003. Stoch. Model. Appl. Probab. 35 (2003).
B. Lapeyre and J. Lelong, A framework for adaptive Monte–Carlo procedures. Monte Carlo Methods Appl. (2011).
J. Lelong, Almost sure convergence of randomly truncated stochastic agorithms under verifiable conditions. Stat. Probab. Lett. 78 (2009).
Lemaire, V. and Pagès, G., Unconstrained Recursive Importance Sampling. Ann. Appl. Probab. 20 (2010) 1029–1067. Google Scholar
Pelletier, M., Weak convergence rates for stochastic approximation with application to multiple targets and simulated annealing. Ann. Appl. Probab. 8 (1998) 10–44. Google Scholar
Robbins, H. and Monro, S., A stochastic approximation method. Ann. Math. Statistics 22 (1951) 400–407. Google Scholar