High-dimensional central limit theorems by Stein’s method

We obtain explicit error bounds for the d-dimensional normal approximation on hyperrectangles a random vector that has Stein kernel, or admits an exchangeable pair coupling, is nonlinear statistic of independent variables sum n locally dependent vectors. assume approximating distribution nonsingular covariance matrix. The vanish even when dimension d much larger than sample size n. prove our main results using approach Götze (1991) in Stein’s method, together with modifications estimate Anderson, Hall and Titterington (1998) smoothing inequality Bhattacharya Rao (1976). For sums identically distributed isotropic vectors having log-concave density, we bound optimal up to logn factor. also discuss application multiple Wiener–Itô integrals.