Stein's Method and Self-Normalized Berry–Esseen Inequality

Abstract

The standard method to prove central limit theorems and Berry—Esseen inequalities is based on characteristic functions, as shown in Sect. 2.3. A different method to derive normal approximations was introduced by Stein (1972). Stein's method works well not only for independent random variables but also for dependent ones. It can also be applied to many other probability approximations, notably to Poisson, Poisson process, compound Poisson and binomial approximations. In this chapter we give an overview of the use of Stein's method for normal approximations. We start with basic results on the Stein equations and their solutions and then prove several classical limit theorems and the Berry—Esseen inequality for self-normalized sums.