# Fundamentals of Stein's method

@article{Ross2011FundamentalsOS, title={Fundamentals of Stein's method}, author={Nathan Ross}, journal={Probability Surveys}, year={2011}, volume={8}, pages={210-293} }

This survey article discusses the main concepts and techniques of Stein's method for distributional approximation by the normal, Poisson, exponential, and geometric distributions, and also its relation to concentration of measure inequalities. The material is presented at a level accessible to beginning raduate students studying probability with the main emphasis on the themes that are common to these topics and also to much of the Stein's method literature.

## 329 Citations

A short survey of Stein's method

- Mathematics
- 2014

Stein's method is a powerful technique for proving central limit theorems in probability theory when more straightforward approaches cannot be implemented easily. This article begins with a survey of…

Stein's method for Borel approximation

- Mathematics
- 2019

We develop the tools necessary to use Stein's method for approximation by a Borel distribution. Two applications of these results are discussed. One is an explicit error bound in the approximation of…

The Stein Method

- Mathematics
- 2016

Methodological aspects of the Stein method are exceptionally well discussed in the literature, and for more advanced applications the reader is advised to consult the books and papers referenced at…

Stein's Method and the Laplace Distribution

- Mathematics
- 2012

Using Stein's method techniques, we develop a framework which allows one to bound the error terms arising from approximation by the Laplace distribution and apply it to the study of random sums of…

Archimedes, Gauss, and Stein

- Physics, Mathematics
- 2012

We discuss a characterization of the centered Gaussian distribution which can be read from results of Archimedes and Maxwell, and relate it to Charles Stein's well-known characterization of the same…

Stein’s method and the distribution of the product of zero mean correlated normal random variables

- MathematicsCommunications in Statistics - Theory and Methods
- 2019

Abstract Over the last 80 years there has been much interest in the problem of finding an explicit formula for the probability density function of two zero mean correlated normal random variables.…

Stein’s method via induction

- MathematicsElectronic Journal of Probability
- 2020

Applying an inductive technique for Stein and zero bias couplings yields Berry-Esseen theorems for normal approximation with optimal rates in the Kolmogorov metric for two new examples. The…

Approximating by convolution of the normal and compound Poisson laws via Stein’s method

- Mathematics
- 2018

We apply Stein’s method to estimate the closeness of the mixtures of distributions to the convolution of the normal and Poisson laws. To derive the Stein operator, we use a moment generating…

Stein's Method Meets Computational Statistics: A Review of Some Recent Developments

- Mathematics
- 2021

Stein’s method compares probability distributions through the study of a class of linear operators called Stein operators. While mainly studied in probability and used to underpin theoretical…

Stein's method for comparison of univariate distributions

- Mathematics
- 2014

We propose a new general version of Stein's method for univariate distributions. In particular we propose a canonical definition of the Stein operator of a probability distribution {which is based on…

## References

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Stein's Method: Expository Lectures and Applications

- Mathematics
- 2004

A review of Stein’s method applied to the case of discrete random variables and attempt to complete one of Stein's open problems, that of providing a discrete version for chapter 6 of his book.

Normal Approximation by Stein ’ s Method

- Mathematics
- 2003

The aim of this paper is to give an overview of Stein’s method, which has turned out to be a powerful tool for estimating the error in normal, Poisson and other approximations, especially for sums of…

Stein's method for geometric approximation

- Mathematics
- 1996

The Stein–Chen method for Poisson approximation is adapted to the setting of the geometric distribution. This yields a convenient method for assessing the accuracy of the geometric approximation to…

Exponential Approximation by Stein's Method and Spectral Graph Theory

- Mathematics
- 2006

General Berry-Esseen bounds are developed for the exponential distri- bution using Stein's method and a new concentration inequality approach. As an application, a sharp error term is obtained for…

An Introduction to Stein's Method

- Mathematics
- 2005

A common theme in probability theory is the approximation of complicated probability distributions by simpler ones, the central limit theorem being a classical example. Stein's method is a tool which…

Stein ’ s method for concentration inequalities

- Mathematics
- 2007

We introduce a version of Stein’s method for proving concentration and moment inequalities in problems with dependence. Simple illustrative examples from combinatorics, physics, and mathematical…

Poisson Approximation and the Chen-Stein Method

- Mathematics
- 1990

The Chen-Stein method of Poisson approximation is a powerful tool for computing an error bound when approximating probabilities using the Poisson distribution. In many cases, this bound may be given…

Applications of Stein's method for concentration inequalities

- Mathematics
- 2010

Stein’s method for concentration inequalities was introduced to prove concentration of measure in problems involving complex dependencies such as random permutations and Gibbs measures. In this…

Stein couplings for normal approximation

- Mathematics
- 2010

In this article we propose a general framework for normal approximation using Stein's method. We introduce the new concept of Stein couplings and we show that it lies at the heart of popular…

New rates for exponential approximation and the theorems of R\'{e}nyi and Yaglom

- Mathematics
- 2009

We introduce two abstract theorems that reduce a variety of complex exponential distributional approximation problems to the construction of couplings. These are applied to obtain rates of…