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

09/30/2019 ∙ by Dali Liu, et al. ∙ 0

Since Stein presented his ideas in the seminal paper 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 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 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 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 Δ=W-W' , Rinott and Rotar rr, Shao and Su ss obtained the Berry-Esseen bound of the normal approximation when Δ is bounded. If Δ is unbounded, Chen and Shao 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 ss , Chen and Shao cs2. Recently, Shao and Zhang sz made a big break for unbounded Δ without using the concentration inequality. They obtained a simple bound as seen from the following result.

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