Asymptotic normality of randomly truncated stochastic algorithms
ESAIM: Probability and Statistics, Tome 17 (2013), pp. 105-119.

We study the convergence rate of randomly truncated stochastic algorithms, which consist in the truncation of the standard Robbins-Monro procedure on an increasing sequence of compact sets. Such a truncation is often required in practice to ensure convergence when standard 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 local assumptions on the expected-value function, which are fairly easy to check in practice.

DOI : 10.1051/ps/2011110
Classification : 62L20, 60F05, 62F12
Mots-clés : stochastic approximation, central limit theorem, randomly truncated stochastic algorithms, martingale arrays
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     url = {http://www.numdam.org/articles/10.1051/ps/2011110/}
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Lelong, Jérôme. Asymptotic normality of randomly truncated stochastic algorithms. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 105-119. doi : 10.1051/ps/2011110. http://www.numdam.org/articles/10.1051/ps/2011110/

[1] B. Arouna, Adaptative Monte Carlo method, a variance reduction technique. Monte Carlo Methods Appl. 10 (2004) 1-24. | MR | Zbl

[2] 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. | MR | Zbl

[3] C. Bouton, Approximation Gaussienne d'algorithmes stochastiques à dynamique Markovienne. Ph.D. Thesis, Université Pierre et Marie Curie, Paris 6 (1985). | Zbl

[4] R. Buche and H.J. Kushner, Rate of convergence for constrained stochastic approximation algorithms. SIAM J. Control Optim. 40 (2001) 1011-1041 (electronic). | MR | Zbl

[5] H.-F. Chen, Stochastic approximation and its applications, Kluwer Academic Publishers, Dordrecht. Nonconvex Optim. Appl. 64 (2002). | MR | Zbl

[6] H. Chen and Y. Zhu, Stochastic Approximation Procedure with randomly varying truncations. Scientia Sinica Series (1986). | MR | Zbl

[7] B. Delyon, General results on the convergence of stochastic algorithms. IEEE Trans. Automat. Contr. 41 (1996) 1245-1255. | MR | Zbl

[8] M. Duflo, Algorithmes stochastiques (Mathématiques et Applications). Springer (1996). | MR | Zbl

[9] M. Duflo, Random Iterative Models. Springer-Verlag Berlin and New York (1997). | MR | Zbl

[10] 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). | MR | Zbl

[11] B. Lapeyre and J. Lelong, A framework for adaptive Monte-Carlo procedures. Monte Carlo Methods Appl. (2011). | Zbl

[12] J. Lelong, Almost sure convergence of randomly truncated stochastic agorithms under verifiable conditions. Stat. Probab. Lett. 78 (2009). | MR | Zbl

[13] V. Lemaire and G. Pagès, Unconstrained Recursive Importance Sampling. Ann. Appl. Probab. 20 (2010) 1029-1067. | MR | Zbl

[14] M. Pelletier, Weak convergence rates for stochastic approximation with application to multiple targets and simulated annealing. Ann. Appl. Probab. 8 (1998) 10-44. | MR | Zbl

[15] H. Robbins and S. Monro, A stochastic approximation method. Ann. Math. Statistics 22 (1951) 400-407. | MR | Zbl

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