Stein’s method for concentration inequalities

Abstract

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.

An Introduction to Matrix Concentration Inequalities

“…Recently, Lester Mackey and the author, in collaboration with Daniel Paulin and several other researchers [118,143], have developed another framework for establishing matrix concentration. This approach extends a scalar argument, introduced by Chatterjee [38,39], that depends on exchangeable pairs and Markov chain couplings. The method of exchangeable pairs delivers both exponential concentration inequalities and polynomial moment inequalities for random matrices, and it can reproduce many of the bounds mentioned above.…”

Matrix Concentration Inequalities via the Method of Exchangeable Pairs

“…Our work is based on Chatterjee's technique for developing scalar concentration inequalities [6,7] via Stein's method of exchangeable pairs [47]. We extend this argument to the matrix setting, where we use it to establish exponential concentration bounds (Theorems 4.1 and 5.1) and polynomial moment inequalities (Theorem 7.1) for the spectral norm of a random matrix.…”

An Optimization Method of Deterministic Measurement Matrix in Distributed Compressed Video Sensing

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