Multivariate, combinatorial and discretized normal approximations by Stein's method

Abstract

This thesis consists of three topics in normal approximation by Stein's method described as follows. 1. Multivariate normal approximation: Under the setting of Stein coupling, we obtain bounds on non-smooth function distances between the distribution of a sum of dependent random vectors and the multivariate normal distribution using the recursive approach. By extending the concentration inequality approach to the multivariate setting, a multivariate normal approximation theorem on convex sets is also proved for sums of independent random vectors. 2. Combinatorial central limit theorem: Using the concentration inequality approach, a Berry-Esseen bound is obtained for a combinatorial central limit theorem where the components of the matrix are assumed to be independent random variables. 3. Discretized normal approximation: Under the setting of Stein coupling, the distributions of sums of dependent integer-valued random variables are approximated by discretized normal distributions in total variation.