Stein’s method in high dimensions with applications

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\to\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.