A non-uniform Berry-Esseen bound via Stein's method
Let \(X_1,X_2,\ldots,X_n\) be independent random variables with zero means and suppose \(W=\sum_{i=1}^nX_i\) has unit variance. The main result (Theorem~2.2) is that there exists an absolute constant \(C\) such that for all real numbers \(x\), \[ |F(x)-\Phi(x)|\leq C\sum_{i=1}^n\left\{\frac{E X_i^2I(|X_i|>1+|x|)}{(1+|x|)^2} +\frac{E|X_i|^3I(|X_i|\leq 1+|x|)}{(1+|x|)^3}\right\},\tag \(*\) \] where \(F\) is the distribution function of \(W\) and \(\Phi\) is the standard normal distribution function. The approach is via Stein's method and a concentration inequality. The method is first illustrated by deriving a uniform bound for \(\sup_{-\infty<x<\infty}|F(x)-\Phi(x)|\), namely the right hand side of \((*)\) with \(x=0\) and \(C=4.1\). This bound, with \(C=6\), was originally obtained by \textit{W. Feller} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 10, 261-268 (1968; Zbl 0167.17304)].
scientific article; zbMATH DE number 1649182