Stein's method for normal approximation

Abstract

Stein’s method originated in 1972 in a paper in the Proceedings of the Sixth Berkeley Symposium. In that paper, he introduced the method in order to determine the accuracy of the normal approximation to the distribution of a sum of dependent random variables satisfying a mixing condition. Since then, many developments have taken place, both in extending the method beyond normal approximation and in applying the method to problems in other areas. In these lecture notes, we focus on univariate normal approximation, with our main emphasis on his approach exploiting an a priori estimate of the concentration function. We begin with a general explanation of Stein’s method as applied to the normal distribution. We then go on to consider expectations of smooth functions, first for sums of independent and locally dependent random variables, and then in the more general setting of exchangeable pairs. The later sections are devoted to the use of concentration inequalities, in obtaining both uniform and non-uniform Berry–Esseen bounds for independent random variables. A number of applications are also discussed.