An almost sure central limit theorem for stochastic approximation algorithms

Journal of Multivariate Analysis

Volumn 71, Issue 1, 1999, Pages 76-93

  Pelletier, Mariane ^a  

^a UNIVERSITÉ GUSTAVE EIFFEL   (France)

Author keywords

Almost sure invariance principles; Central limit theorem; Stochastic approximation algorithms

Indexed keywords

EID: 0013306106     PISSN: 0047259X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jmva.1999.1830     Document Type: Article

References (20)

1. A. Benveniste, M. Métivier, and P. Priouret, "Adaptive Algorithms and Stochastic Approximation," Springer-Verlag, New York/Berlin, 1990.
2. G. A. Brosamler, An almost everywhere central limit theorem, Math. Proc. Cambridge Philos. Soc. 104 (1988), 213-246.
3. F. Chaabane, Version forte du théorème de la limite centrale fonctionnel pour les martingales, C. R. Acad. Sci. Paris Sér. I 323 (1996), 195-198.
4. F. Chaabane, F. Maaouia, and A. Touati, Versions fortes associées aux théorèmes limite en loi pour les martingales vectorielles, preprint Faculté des sciences de Bizerte, Tunisie, 1996.
5. H. F. Chen, L. Guo, and A. J. Gao, Convergence and robustness of the Robbins-Monro algorithm truncated at randomly varying bounds, Stochastic Process. Appl. (1988), 217-231.
6. D. Dacunha Castelle and M. Duflo, "Probability and Statistics," Vol. 1, Springer-Verlag, New York/Berlin, 1986.
7. M. Duflo, "Algorithmes stochastiques," Collection Mathématiques et Applications, Springer-Verlag, New York/Berlin, 1996.
8. P. Dupuis and H. J. Kushner, Stochastic approximation and large deviations: Upper bounds and w.p. 1 convergence, SIAM J. Control Optim. 27 (1989), 1108-1135.
9. W. Feller, "An Introduction to Probability Theory and Its Applications," 3rd ed., Vol. 2, Wiley, New York, 1968.
10. H. J. Kushner and D. S. Clark, Stochastic approximation for constrained and unconstrained systems, in "Appl. Math. Sci.," Vol. 26, Springer-Verlag, New York/Berlin, 1978.
11. M. Lacey and W. Philipp, A note on the almost sure central limit theorem, Statist. Probab. Lett. 9 (1990), 201-205.
12. T. Z. Lai and C. Z. Wei, A note on martingale difference sequences satisfying the local Marcinkiewicz-Zigmund condition, Bull. Inst. Math. Acad. Sinica 11 (1983), 1-13.
13. L. Ljung, G. Pflug, and H. Walk, "Stochastic Approximation and Optimization of Random Systems," Birkhäuser, Basel, 1992.
14. M. B. Nevelson and R. Z. Has'minskii, "Stochastic Approximation and Recursive Estimations," Transl. Math. Monogr., Vol. 47, Am. Math. Soc., Providence, 1973.
15. M. Pelletier, Weak convergence rates for stochastic approximation with application to multiple targets and simulated annealing, Ann. Appl. Probab. 8 (1998), 10-44.
16. M. Pelletier, On the almost sure asymptotic behaviour of stochastic algorithms, Stochastic Process. Appl. 78 (1998), 217-244.
17. B. Rodzik and Z. Rychlik, An almost sure central limit theorem for independent random variables, Ann. Inst. Henri Poincaré 30 (1994), 1-11.
18. P. Schatte, On strong versions of central limit theorem, Math. Nachr. 137 (1988), 249-256.
19. A. Touati, Sur les versions fortes du théorème de la limite centrale, preprint, No. 23, Université de Marne-la-vallée, 1995.
20. Y. Zhu, Asymptotic normality for a vector stochastic difference equation with applications in stochastic approximation, J. Multivariate Anal. 57 (1996), 101-118.