Normal Approximation by Stein’s Method

✍ Scribed by Louis H.Y. Chen, Larry Goldstein, Qi-Man Shao (auth.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 2011
• Tongue: English
• Leaves: 418
• Series: Probability and Its Applications
• Edition: 1
• Category: Library

✦ Synopsis

Since its introduction in 1972, Stein’s method has offered a completely novel way of evaluating the quality of normal approximations. Through its characterizing equation approach, it is able to provide approximation error bounds in a wide variety of situations, even in the presence of complicated dependence. Use of the method thus opens the door to the analysis of random phenomena arising in areas including statistics, physics, and molecular biology.

Though Stein's method for normal approximation is now mature, the literature has so far lacked a complete self contained treatment. This volume contains thorough coverage of the method’s fundamentals, includes a large number of recent developments in both theory and applications, and will help accelerate the appreciation, understanding, and use of Stein's method by providing the reader with the tools needed to apply it in new situations. It addresses researchers as well as graduate students in Probability, Statistics and Combinatorics.

✦ Table of Contents

Front Matter....Pages I-XII
Introduction....Pages 1-12
Fundamentals of Stein’s Method....Pages 13-44
Berry–Esseen Bounds for Independent Random Variables....Pages 45-62
L 1 Bounds....Pages 63-145
L ∞ by Bounded Couplings....Pages 147-166
L ∞ : Applications....Pages 167-220
Discretized Normal Approximation....Pages 221-232
Non-uniform Bounds for Independent Random Variables....Pages 233-244
Uniform and Non-uniform Bounds Under Local Dependence....Pages 245-255
Uniform and Non-uniform Bounds for Non-linear Statistics....Pages 257-291
Moderate Deviations....Pages 293-312
Multivariate Normal Approximation....Pages 313-341
Non-normal Approximation....Pages 343-369
Group Characters and Malliavin Calculus....Pages 371-388
Back Matter....Pages 389-405

✦ Subjects

Probability Theory and Stochastic Processes