Stein’s method in high dimensions with applications

Abstract

Let $h$ be a three times partially differentiable function on $R^n$, let $X=(X_1,\dots,X_n)$ be a collection of real-valued random variables and let $Z=(Z_1,\dots,Z_n)$ be a multivariate Gaussian vector. In this article, we develop Stein's method to give error bounds on the difference $E h(X) - E h(Z)$ 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.