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.
Keywords
- Independent Random Variable
- Lipschitz Function
- Lipschitz Continuous Function
- Random Index
- Concentration Inequality
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© 2011 Springer-Verlag Berlin Heidelberg
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Chen, L.H.Y., Goldstein, L., Shao, QM. (2011). Berry–Esseen Bounds for Independent Random Variables. In: Normal Approximation by Stein’s Method. Probability and Its Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15007-4_3
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DOI: https://doi.org/10.1007/978-3-642-15007-4_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15006-7
Online ISBN: 978-3-642-15007-4
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