Normal Approximation by Stein’s Method (Probability and Its Applications)

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. 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