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Normal Approximation and Asymptotic Expansions

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By Rabi N. Bhattacharya and R. Ranga Rao


2010 / xxii + 316 / Softcover / ISBN: 978-0-898718-97-3 / List Price $82.00 / SIAM Member Price $57.40 / Order Code CL64

Keywords: Berry-Essen bounds, asymptotic expansions

Although Normal Approximation and Asymptotic Expansions was first published in 1976, it has gained new significance and renewed interest among statisticians due to the developments of modern statistical techniques such as the bootstrap, the efficacy of which can be ascertained by asymptotic expansions.

This also is the only book containing a detailed treatment of various refinements of the multivariate central limit theorem (CLT), including Berry–Essen-type error bounds for probabilities of general classes of functions and sets, and asymptotic expansions for both lattice and non-lattice distributions. With meticulous care, the authors develop necessary background on

  • weak convergence theory, 
  • Fourier analysis, 
  • geometry of convex sets, and
  • the relationship between lattice random vectors and discrete subgroups of Rk.

The formalism developed in the book has been used in the extension of the theory by Goetze and Hipp to sums of weakly dependent random vectors.

This edition of the book includes a new chapter that provides an application of Stein’s method of approximation to the multivariate CLT.

The book is appropriate for graduate students of probability and statistics as well as researchers in these and other fields whose work involves the asymptotic theory of statistics.

Preface to the Classics Edition;
Chapter 1: Weak Convergence of Probability Measures and Uniformity Classes;
Chapter 2: Fourier Transforms and Expansions of Characteristic Functions;
Chapter 3: Bounds for Errors of Normal Approximation;
Chapter 4: Asymptotic ExpansionsÑNonlattice Distributions;
Chapter 5: Asymptotic ExpansionsÑLattice Distributions;
Chapter 6: Two Recent Improvements;
Chapter 7: An Application of SteinÕs Method;
Appendix A.1: Random Vectors and Independence;
Appendix A.2: Functions of Bounded Variation and Distribution Functions;
Appendix A.3: Absolutely Continuous, Singular, and Discrete Probability Measures;
Appendix A.4: The Euler-MacLaurin Sum Formula for Functions of Several Variables

About the Authors
Rabi N. Bhattacharya is a Professor of Mathematics at the University of Arizona. He is an IMS Fellow, a member of the AMS, and a recipient of the Humboldt Prize and a Guggenheim Fellowship. He has co-authored a number of graduate texts and research monographs, including Stochastic Processes with Applications (with E.C. Waymire), which has also been republished in the SIAM Classics in Applied Mathematics series.

R. Ranga Rao is an Emeritus Professor of Mathematics at the University of Illinois. He has held a number of visiting professorships, including several at the Tata Institute of Fundamental Research in India. He is also a member of the AMS.

ISBN: 9780898718973

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