$\mathrm{L}_\infty$-Bound for Asymptotic Normality of Weakly Dependent Summands Using Stein's Result

Abstract

Let $\{X_n\}$ be a strictly stationary process satisfying some mixing conditions, including $\phi$-mixing condition. It is the aim of the present paper to give, using a slight modification of Stein's result, a rate $O(n^{-1/2} \log n)$ of the normal approximation for a sum $S_n = X_1 + \cdots + X_n$.