High-dimensional Central Limit Theorems by Stein's Method

Understand

We obtain explicit error bounds 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 the approach of G\"otze (1991) in Stein's method, together with modifications of an estimate of Anderson, Hall and Titterington (1998) and a smoothing inequality of Bhattacharya and Rao (1976).