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 $Eh(X)-Eh(Z)$ 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.

22 pages, published version

References in corpus (5)

• A simple invariance theorem
• Bulk universality for generalized Wigner matrices
• Stein couplings for normal approximation
• Stein's method for dependent random variables occurring in Statistical Mechanics
• A GUE Central Limit Theorem and Universality of Directed First and Last Passage Site Percolation

Cited by in corpus (16)

• Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors
• Uniform Post Selection Inference for LAD Regression and Other Z-estimation problems
• A multivariate Berry--Esseen theorem with explicit constants
• A short survey of Stein's method
• Bootstrapping High Dimensional Time Series
• On Stein's Method for Multivariate Self-Decomposable Laws
• A multivariate central limit theorem for Lipschitz and smooth test functions
• Limit theorems for empirical Fréchet means of independent and non-identically distributed manifold-valued random variables
• On Stein's Method for Multivariate Self-Decomposable Laws With Finite First Moment
• Stein's method for dynamical systems
• The Bootstrap for Network Dependent Processes
• Bootstrapping ${\ell }_p$-Statistics in High Dimensions
• Central limit theorems with a rate of convergence for sequences of transformations
• Estimating the Penalty Level of ${\ell }_1$-minimization via Two Gaussian Approximation Methods
• The critical disordered pinning measure
• Simulation and analytical approach to the identification of significant factors