A0271
Title: High-dimensional central limit theorems by Stein's method
Authors: Xiao Fang - The Chinese University of Hong Kong (Hong Kong) [presenting]
Abstract: Explicit error bounds are obtained for the $d$-dimensional normal approximation on hyperrectangles for a random vector that has a Stein kernel, or admits an exchangeable pair coupling, or is a non-linear statistic of independent random variables or a sum of $n$ locally dependent random vectors. We assume the approximating normal distribution has a non-singular covariance matrix. The error bounds vanish even when the dimension $d$ is much larger than the sample size $n$. We prove our main results using a previous approach in Stein's method, together with modifications of a given estimate and a smoothing inequality. For sums of $n$ independent and identically distributed isotropic random vectors having a log-concave density, we obtain an error bound that is optimal up to a $\log n$ factor. We also discuss an application to multiple Wiener-It\^{o}integrals.
