Stein’s method: expository lectures and applications. Papers from the workshop on Stein’s method, Stanford, CA, USA, 1998.

*(English)*Zbl 1079.62024
Institute of Mathematical Statistics Lecture Notes - Monograph Series 46. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 0-940600-62-5/pbk). vi, 139 p., open access (2004).

This monograph is a collection of expository talks presented at a workshop on Stein’s method at Stanford’s department of Statistics in 1998. Originally proposed by Charles Stein in 1976, Stein’s method has become a powerful alternative to classical methods involving Fourier analysis or moments in proving limit theorems for Poisson, normal and other classical approximations. The papers in this monograph illustrate Stein’s method through a series of novel applications and examples.

The first paper, written by Stein himself with colleagues, presents the application of Stein’s method to the improvement of the accuracy of standard Monte Carlo simulation procedures and estimates of approximation errors. The second paper by Persi Diaconis uses Stein’s method to derive rates of convergence of some simple Markov chains to their stationary distributions and compares this method with several alternatives in this simple setting, where the applications to discrete random variables including birth and death chains are reviewed in the third paper by Susan Holmes. In the fourth paper, Jason Fulman derives a central limit theorem for the number of descents or inversions of permutations on a symmetric group by using Stein’s method and non-reversible Markov chains. Mark Huber and Gesine Reinert discuss in the fifth paper the exact sampling and approximations of the stationary distributions for the antivoter model in random graph theory. Stein’s method is used to bound the total variation distance between the stationary distribution for the antivoter model on a multipartite graph and the stationary distribution on the complete graph. In the last paper, the techniques based on Stein’s method for empirical processes are employed to give new proofs for many known results about the convergence in law of bootstrap distributions to the true distribution of smooth statistics and to suggest a new bootstrap procedure which works under uniform local dependence.

In summary, this monograph summarizes some recent results on the applications of Stein’s method which should be of interest and useful to researchers who are considering the applications of Stein’s method to other situations or are interested in new results in the diverse areas covered in this monograph.

The first paper, written by Stein himself with colleagues, presents the application of Stein’s method to the improvement of the accuracy of standard Monte Carlo simulation procedures and estimates of approximation errors. The second paper by Persi Diaconis uses Stein’s method to derive rates of convergence of some simple Markov chains to their stationary distributions and compares this method with several alternatives in this simple setting, where the applications to discrete random variables including birth and death chains are reviewed in the third paper by Susan Holmes. In the fourth paper, Jason Fulman derives a central limit theorem for the number of descents or inversions of permutations on a symmetric group by using Stein’s method and non-reversible Markov chains. Mark Huber and Gesine Reinert discuss in the fifth paper the exact sampling and approximations of the stationary distributions for the antivoter model in random graph theory. Stein’s method is used to bound the total variation distance between the stationary distribution for the antivoter model on a multipartite graph and the stationary distribution on the complete graph. In the last paper, the techniques based on Stein’s method for empirical processes are employed to give new proofs for many known results about the convergence in law of bootstrap distributions to the true distribution of smooth statistics and to suggest a new bootstrap procedure which works under uniform local dependence.

In summary, this monograph summarizes some recent results on the applications of Stein’s method which should be of interest and useful to researchers who are considering the applications of Stein’s method to other situations or are interested in new results in the diverse areas covered in this monograph.

Reviewer: Dongsheng Tu (Kingston)

##### MSC:

62E20 | Asymptotic distribution theory in statistics |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60F05 | Central limit and other weak theorems |

60J22 | Computational methods in Markov chains |

60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |

05C80 | Random graphs (graph-theoretic aspects) |

62G09 | Nonparametric statistical resampling methods |

65C05 | Monte Carlo methods |

60B15 | Probability measures on groups or semigroups, Fourier transforms, factorization |