Stein's Method, Self-normalized Limit Theory and Applications

Stein's method is a powerful tool in estimating accuracy of various probability approximations. It works for both independent and dependent random variables. It works for normal approximation and also for non-normal approximation. The method has been successfully applied to study the absolute error of approximations and the relative error as well. In contrast to the classical limit theorems, the self-normalized limit theorems require no moment assumptions or much less moment assumptions. This paper is devoted to the latest developments on Stein's method and self-normalized limit theory. Starting with a brief introduction on Stein's method, recent results are summarized on normal approximation for smooth functions and Berry-Esseen type bounds, Cramer type moderate deviations under a general framework of the Stein identity, non normal approximation via exchangeable pairs, and a randomized exponential concentration inequality. For self-normalized limit theory, the focus will be on a general self-normalized moderate deviation, the self-normalized saddlepoint approximation without any moment assumption, Cramer type moderate deviations for maximum of self-normalized sums and for Studentized U-statistics. Applications to the false discovery rate in simultaneous tests as well as some open questions will also be discussed.