Stein's method for diffusion approximations

Barbour, A. D.

doi:10.1007/bf01197887

Cited by 189 publications

(282 citation statements)

References 8 publications

Supporting

Mentioning

281

Contrasting

Order By: Relevance

“…There are other variants of Stein's method, most notably the generator method of Andrew Barbour [4], the dependency graph approach introduced by Chen [15] and Baldi and Rinott [3] and popularized by Arratia, Goldstein and Gordon [2], the size-biased coupling method of Barbour, Holst and Janson [5], and the zero-biased coupling method due to Goldstein and Reinert [23]. The recent applications to algebraic problems by Jason Fulman [20,21], and the quest for Berry-Esseen bounds by Rinott and Rotar [34] and Shao and Su [35] are also worthy of note.…”

Section: Proposition 14 Let R Be the Maximum Degree Of The Dependenmentioning

confidence: 99%

Stein’s method for concentration inequalities

Chatterjee

2006

Probab. Theory Relat. Fields

148

131

View full text Add to dashboard Cite

Abstract. 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 statistics are provided. Introduction and resultsStein's method was introduced by Charles Stein [38] in the context of normal approximation for sums of dependent random variables. Stein's version of his method, best known as the "method of exchangeable pairs", attained maturity in his later work [39]. A reasonably large literature has developed around the subject, but it has almost exclusively developed as a method of proving distributional convergence with error bounds. Stein's attempts at getting large deviations in [39] did not, unfortunately, prove fruitful. Some progress for sums of dependent random variables was made by Raič [33]. A general version of Stein's method for concentration inequalities was introduced for the first time in the Ph.D. thesis [11] of the present author. The purpose of this paper is to explain the theory developed in [11] via examples. Another application is in [12].This section is organized as follows: First, we give three examples, followed by the main abstract theorem; finally, towards the end of the section, we present very condensed overviews of Stein's method, concentration of measure, and the related literature. Proofs are in section 2.1.1. A generalized matching problem. Let {a ij } be an n × n array of real numbers. Let π be chosen uniformly at random from the set of all permutations of {1, . . . , n}, and let X = n i=1 a iπ(i) . This class of random variables was first studied by Hoeffding [24], who proved that they are approximately normally distributed under certain conditions. It is easy to see that various well-studied functions of random permutations, like the number of fixed points, the sum of a random sample picked without replacement from a finite population, and the function i |i−π(i)| (known as Spearman's footrule [16]), are all instances of Hoeffding's statistic.2000 Mathematics Subject Classification. 60E15; 60C05; 60K35; 82C22.

show abstract

Section: Proposition 14 Let R Be the Maximum Degree Of The Dependenmentioning

confidence: 99%

Stein’s method for concentration inequalities

Chatterjee

2006

Probab. Theory Relat. Fields

148

131

View full text Add to dashboard Cite

show abstract

“…Normal approximations through the use of identity (1), first provided by Stein [14], have since been obtained by other authors using variations in abundance (see e.g. [15], [5], [13] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Zero biasing in one and higher dimensions, and applications

Goldstein¹,

Reinert²

2005

Stein's Method and Applications

View full text Add to dashboard Cite

Given any mean zero, finite variance σ 2 random variable W , there exists a unique distribution on a variable W * such that EW f (W ) = σ 2 Ef (W * ) for all absolutely continuous functions f for which these expectations exist. This distributional 'zero bias' transformation of W to W * , of which the normal is the unique fixed point, was introduced in [9] to obtain bounds in normal approximations. After providing some background on the zero bias transformation in one dimension, we extend its definition to higher dimension and illustrate with an application to the normal approximation of sums of vectors obtained by simple random sampling.

show abstract

“…A number of techniques for carrying out this step are available in the literature, e.g. exchangeable pairs [27], diffusion generators [4], dependency graphs [2,3], size bias couplings [16], zero bias couplings [15], couplings for Poisson approximation [5,11], specialized procedures like [19,22,23], and some recent advances [7][8][9][10]. Incidentally, Stein's method was applied to solve a problem in the interface of statistics and spin glasses in [6].…”

Section: By the Definition Of F It Follows Thatmentioning

confidence: 99%

Spin glasses and Stein’s method

Chatterjee

2009

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

We introduce some applications of Stein's method in the high temperature analysis of spin glasses. Stein's method allows the direct analysis of the Gibbs measure without having to create a cavity. Another advantage is that it gives limit theorems with total variation error bounds, although the bounds can be suboptimal. A surprising byproduct of our analysis is a relatively transparent explanation of the Thouless-Anderson-Palmer system of equations. Along the way, we develop Stein's method for mixtures of two Gaussian densities.

show abstract

Stein's method for diffusion approximations

Cited by 189 publications

References 8 publications

Stein’s method for concentration inequalities

Stein’s method for concentration inequalities

Zero biasing in one and higher dimensions, and applications

Spin glasses and Stein’s method

Contact Info

Product

Resources

About