Stein's method in high dimensions with applications

Abstract

Let h be a three times partially differentiable function on Rn, let X = (X1, +⋯, Xn) be a collection of real-valued random variables and let Z = (Z1, +⋯, Zn) be a multivariate Gaussian vector. In this article, we develop Stein's method to give error bounds on the difference Eh(X) - Eh(Z) in cases where the coordinates of X are not necessarily independent, focusing on the high dimensional case n→∞. 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. © 2013 Association des Publications de l'Institut Henri Poincaré.