Berry–Esseen Bounds for Independent Random Variables

Abstract

Chapter 3 focuses on independent random variables. The main goal of the chapter is to demonstrate a version of the classical Berry–Esseen theorem using Stein’s method. Along the way, techniques are developed for obtaining L ¹ bounds, and the Lindeberg central limit theorem is proved as well. The Berry–Esseen theorem is first demonstrated for the case where the random variables are bounded. The boundedness condition is then relaxed in two ways, first by concentration inequalities, then by induction. This chapter concludes with a lower bound for the Berry–Esseen inequality. As seen in the chapter dependency diagram that follows, Chaps. 2 and 3 form much of the basis of this book.