We introduce different ways of modeling the dependency of the input noise of stochastic algorithms. We are aimed to prove that such innovations allow us to use the ODE (ordinary differential equation) method. Illustrations in the linear regression framework and in the law of the large number for triangular arrays of weighted dependent random variables are also given. We have aimed to provide results easy to check in practice.
La dépendance du bruit d'un algorithme stochastique est modélisée de différentes manières, de sorte que la méthode de l'équation différentielle ordinaire reste applicable. Ces techniques de dépendance faible sont illustrées ici par des applications à un algorithme de régression linéaire et à l'étude de tableaux triangulaires de variables aléatoires pondérées dépendantes. L'objectif est ici d'obtenir des conditions aisément vérifiables en pratique.
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@article{CRMATH_2003__337_7_473_0, author = {Doukhan, Paul and Brandi\`ere, Odile}, title = {Algorithmes stochastiques \`a bruit d\'ependant}, journal = {Comptes Rendus. Math\'ematique}, pages = {473--476}, publisher = {Elsevier}, volume = {337}, number = {7}, year = {2003}, doi = {10.1016/j.crma.2003.07.002}, language = {fr}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2003.07.002/} }
TY - JOUR AU - Doukhan, Paul AU - Brandière, Odile TI - Algorithmes stochastiques à bruit dépendant JO - Comptes Rendus. Mathématique PY - 2003 SP - 473 EP - 476 VL - 337 IS - 7 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2003.07.002/ DO - 10.1016/j.crma.2003.07.002 LA - fr ID - CRMATH_2003__337_7_473_0 ER -
%0 Journal Article %A Doukhan, Paul %A Brandière, Odile %T Algorithmes stochastiques à bruit dépendant %J Comptes Rendus. Mathématique %D 2003 %P 473-476 %V 337 %N 7 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2003.07.002/ %R 10.1016/j.crma.2003.07.002 %G fr %F CRMATH_2003__337_7_473_0
Doukhan, Paul; Brandière, Odile. Algorithmes stochastiques à bruit dépendant. Comptes Rendus. Mathématique, Volume 337 (2003) no. 7, pp. 473-476. doi : 10.1016/j.crma.2003.07.002. http://archive.numdam.org/articles/10.1016/j.crma.2003.07.002/
[1] A dynamical system approach to stochastic approximation, SIAM J. Control Optim., Volume 34 (1996) no. 2, pp. 437-472
[2] Stochastic Approximation and Its Applications, Kluwer Academic, 2002
[3] Some convergence theorems for independent random variables, Ann. Math. Statist., Volume 37 (1966), pp. 1482-1493 36 (4) 1293–1314
[4] P. Dedecker, P. Doukhan, A new covariance inequality and applications, Stochastic Process. Appl. (2002), in press
[5] General convergence result on stochastic approximation, IEEE Trans. Automatic Control, Volume 41 (1996) no. 9
[6] Models inequalities and limit theorems for stationary sequences (Doukhan et al., eds.), Theory and Applications of Long Range Dependence, Birkhäuser, 2002, pp. 43-101
[7] A new weak dependence condition and applications to moment inequalities, Stochastic Process. Appl., Volume 84 (1999), pp. 313-342
[8] Algorithmes Stochastiques, Collect. Math. Appl., 23, Springer, 1996
[9] Stochastic Approximation for Constrained and Uncontrained Systems, Appl. Math. Sci, 26, Springer, 1978
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