Stein's method in high dimensions with applications

Röllin, Adrian

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013), p. 529-549 / Harvested from Numdam

Résumé

Soit $h$ une fonction réelle sur $\mathbb {R}^{n}$ dont les dérivées partielles d’ordre trois existent, soit $X=(X_{1},\ldots ,X_{n})$ un vecteur de variables aléatoire réelles et soit $Z=(Z_{1},\ldots ,Z_{n})$ un vecteur aléatoire Gaussien. Dans cet article, nous établissons par la méthode de Stein une majoration de la différence $\mathbb {E}h(X)-\mathbb {E}h(Z)$ dans le cas où les coordonnées de $X$ ne sont pas nécessairement indépendantes; nous nous concentrons sur le cas de la grande dimension $n\rightarrow \infty $. 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.

Abstract

Let $h$ be a three times partially differentiable function on $\mathbb {R}^{n}$, let $X=(X_{1},\ldots ,X_{n})$ be a collection of real-valued random variables and let $Z=(Z_{1},\ldots ,Z_{n})$ be a multivariate Gaussian vector. In this article, we develop Stein’s method to give error bounds on the difference $\mathbb {E}h(X)-\mathbb {E}h(Z)$ in cases where the coordinates of $X$ are not necessarily independent, focusing on the high dimensional case $n\rightarrow \infty $. 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.

Détail

Publié le : 2013-01-01

MR 3088380 | Zbl 1287.60043

DOI : https://doi.org/10.1214/11-AIHP473
Classification:  60F17,  82B44

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