Stein's Method: Expository Lectures and Applications
Editor: Persi Diaconis
Editor: Susan Holmes
Lecture Notes--Monograph Series, Volume 46
Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2004.
Stein's method is one of the most powerful tools for proving limit theorems with sharp, explicit errors for complex dependent problems. It is curiously hard to grasp how and why it works since it avoids both characteristic functions and higher moments. This book consists of tutorial and survey papers aimed at teaching Stein's method to non specialists. The book provides a self contained development with motivation and full proofs. As a unifying theme, all papers use Stein's approach of exchangeable pairs. In addition to the usual Poisson and Normal approximations the book gives applications to convergence of Markov chains on finite state spaces, to birth and death chains and to empirical process convergence for the bootstrap. One novel feature is the development of Stein's method as an adjunct to simulation via Monte Carlo. Usually, the identities underlying the method give an explicit error term which is bounded. With the present version using exchangeable pairs, the error is given as an explicit expectation for the reversible Markov chain. It can thus be easily simulated to give improvements to classical approximations. The authors of various chapters, Persi Diaconis, Jason Fulman, Gesine Reinert, Susan Holmes, Mark Huber and Charles Stein have used common notation and worked together to achieve a unified treatment.
Subjects:Limit theorems (Probability theory)
Markov processes--Mathematical models
Birth and death processes (Stochastic processes)
Stein, Charles, 1929-
Mathmatical Reviews number (MathSciNet): MR2118599
Zentralblatt Math Identifier: 1079.62024
Copyright © 2004, Institute of Mathematical Statistics.
Use of exchangeable pairs in the analysis of simulations
Charles Stein, Persi Diaconis, Susan Holmes, and Gesine Reinert; 1-25
The stationary distribution in the antivoter model: exact sampling and approximations
Mark Huber, and Gesine Reinert; 75-92