Normal approximation by Stein’s method. (English) Zbl 1213.62027
Probability and Its Applications. Berlin: Springer (ISBN 978-3-642-15006-7/hbk; 978-3-642-26565-5/pbk; 978-3-642-15007-4/ebook). xii, 405 p. (2011).
This book demonstrates the usefulness of Stein’s characteristic equation for the family of normal distributions [C. Stein, Proc. 6th Berkeley Sympos. Math. Statist. Probab., Univ. Calif. 1970, 2, 583–602 (1972; Zbl 0278.60026)] for deriving normal approximation results in a variety of standard and non-standard situations.
After laying out the basics of Stein’s method for proving asymptotic normality (Chapters 1 and 2), the authors are first concerned with reproducing J.W. Lindeberg’s version of the central limit theorem [Math. Zeitschr. 15, 211–225 (1922; JFM 48.0602.04)] as well as Berry-Esseen bounds in the \(L^\infty\) norm (Kolmogorov distance) for normalized sums of independent random variables with Stein’s technique (Chapter 3). Thereafter, the methods of proof are utilized and extended to establish normal approximation results (in Wasserstein (\(L^1\)) and Kolmogorov (\(L^\infty\)) distances) for a collection of rather diverse dependent situations (Chapters 4–6).
The second part of the book (Chapters 7–12) deals with selected advanced topics to which Stein’s idea can be transferred. In particular, discretized normal approximations, non-uniform bounds, nonlinear statistics, moderate deviations, and the multivariate case are studied in detail. The book concludes with material that goes beyond normal approximations on \(\mathbb{R}^p\). Chapter 13 considers non-normal approximations and Chapter 14 deals with group characters and the Malliavin calculus.
After laying out the basics of Stein’s method for proving asymptotic normality (Chapters 1 and 2), the authors are first concerned with reproducing J.W. Lindeberg’s version of the central limit theorem [Math. Zeitschr. 15, 211–225 (1922; JFM 48.0602.04)] as well as Berry-Esseen bounds in the \(L^\infty\) norm (Kolmogorov distance) for normalized sums of independent random variables with Stein’s technique (Chapter 3). Thereafter, the methods of proof are utilized and extended to establish normal approximation results (in Wasserstein (\(L^1\)) and Kolmogorov (\(L^\infty\)) distances) for a collection of rather diverse dependent situations (Chapters 4–6).
The second part of the book (Chapters 7–12) deals with selected advanced topics to which Stein’s idea can be transferred. In particular, discretized normal approximations, non-uniform bounds, nonlinear statistics, moderate deviations, and the multivariate case are studied in detail. The book concludes with material that goes beyond normal approximations on \(\mathbb{R}^p\). Chapter 13 considers non-normal approximations and Chapter 14 deals with group characters and the Malliavin calculus.
Reviewer: Thorsten Dickhaus (Berlin)
MSC:
62E20 | Asymptotic distribution theory in statistics |
62-02 | Research exposition (monographs, survey articles) pertaining to statistics |
60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |
62E17 | Approximations to statistical distributions (nonasymptotic) |
60G50 | Sums of independent random variables; random walks |
60F05 | Central limit and other weak theorems |
65D99 | Numerical approximation and computational geometry (primarily algorithms) |