Soit une fonction réelle sur dont les dérivées partielles d’ordre trois existent, soit un vecteur de variables aléatoire réelles et soit un vecteur aléatoire Gaussien. Dans cet article, nous établissons par la méthode de Stein une majoration de la différence dans le cas où les coordonnées de ne sont pas nécessairement indépendantes; nous nous concentrons sur le cas de la grande dimension . Pour exprimer la structure de dépendance, nous utilisons des couplages de Stein, ce qui permet une large gamme d’applications, par exemple aux modèles d’urnes, au modèles avec dépendance locale, au modèle de Curie-Weiss, etc. Nous présentons aussi des applications au modèle de Sherrington-Kirkpatrick et à la percolation de dernier passage dans des rectangles étroits.
Let be a three times partially differentiable function on , let be a collection of real-valued random variables and let be a multivariate Gaussian vector. In this article, we develop Stein’s method to give error bounds on the difference in cases where the coordinates of are not necessarily independent, focusing on the high dimensional case . In order to express the dependency structure we use Stein couplings, which allows for a broad range of applications, such as classic occupancy, local dependence, Curie-Weiss model, etc. We will also give applications to the Sherrington-Kirkpatrick model and last passage percolation on thin rectangles.
Mots clés : Stein's method, gaussian interpolation, last passage percolation on thin rectangles, Sherrington-Kirkpatrick model, Curie-Weiss model
@article{AIHPB_2013__49_2_529_0, author = {R\"ollin, Adrian}, title = {Stein's method in high dimensions with applications}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {529--549}, publisher = {Gauthier-Villars}, volume = {49}, number = {2}, year = {2013}, doi = {10.1214/11-AIHP473}, mrnumber = {3088380}, zbl = {1287.60043}, language = {en}, url = {http://www.numdam.org/articles/10.1214/11-AIHP473/} }
TY - JOUR AU - Röllin, Adrian TI - Stein's method in high dimensions with applications JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2013 SP - 529 EP - 549 VL - 49 IS - 2 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/11-AIHP473/ DO - 10.1214/11-AIHP473 LA - en ID - AIHPB_2013__49_2_529_0 ER -
%0 Journal Article %A Röllin, Adrian %T Stein's method in high dimensions with applications %J Annales de l'I.H.P. Probabilités et statistiques %D 2013 %P 529-549 %V 49 %N 2 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/11-AIHP473/ %R 10.1214/11-AIHP473 %G en %F AIHPB_2013__49_2_529_0
Röllin, Adrian. Stein's method in high dimensions with applications. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 2, pp. 529-549. doi : 10.1214/11-AIHP473. http://www.numdam.org/articles/10.1214/11-AIHP473/
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