Non-uniform Berry-Esseen Bound by Unbounded Exchangeable Pair Approach.

Since Stein presented his ideas in the seminal paper \cite{s1}, there have been a lot of research activities around Stein's method. Stein's method is a powerful tool to obtain the approximate error of normal and non-normal approximation. The readers are referred to Chatterjee \cite{c} for recent developments of Stein's method. While several works on Stein's method pay attention to the uniform error bounds, Stein's method showed to be powerful on the non-uniform error bounds, too. By Stein's method, Chen and Shao \cite{cls1, cls2} obtained the non-uniform Berry-Esseen bound for independent and locally dependent random variables. The key in their works is the concentration inequality, which also has strong connection with another approach called the \textit{exchangeable pair approach}. The exchangeable pair approach turned out to be an important topic in Stein's method. Let $W$ be the random variable under study. The pair $ (W,W') $ is called an exchangeable pair if $ (W,W')$ and $(W',W) $ share the same distribution. With $ \Delta=W-W' $, Rinott and Rotar \cite{rr}, Shao and Su \cite{ss} obtained the Berry-Esseen bound of the normal approximation when $\Delta$ is bounded. If $\Delta$ is unbounded, Chen and Shao \cite{cs2} provided a Berry-Esseen bound and got the optimal rate for an independence test. The concentration inequality plays a crucial role in previous studies, such as Shao and Su \cite{ss} , Chen and Shao \cite{cs2}. Recently, Shao and Zhang \cite{sz} made a big break for unbounded $\Delta$ without using the concentration inequality. They obtained a simple bound as seen from the following result.