Non-uniform bounds on normal approximation via stein's method and exchangeable pair

Abstract

This dissertation consists of two parts. In the first, we improve the technique of Chuntee and Neammanee (2017) to obtain an exponential non-uniform bound on normal approximation for any random variables whose exchangeable pair can be constructed. Moreover, we give applications of the theorem to obtain non-uniform exponential bounds of normal approximations for sums of bounded independent random variables, the numbers of descents and inversions of permutation and combinatorial central limit theorem for real numbers. In the second part, we construct a new version of the Rosenthal inequality and improve the technique of Simcharoen and Neammanee (2016) to obtain a sharper non-uniform bound on normal approximation of a combinatorial central limit theorem under the condition of finiteness of the third moment.