Berry-Esseen bounds for statistics of weakly dependent samples

Abstract

We prove Berry--Esseen bounds for a general class of asymptotically normal statistics which are functions of $N$ weakly dependent random variables under easily verifiable conditions. In particular, we show, for some $\delta >0$, the validity of the bound $O\left(N^{-1/2}{log}^{\delta }N\right)$ for $U$-statistics, studentized means, functions of sample means, functionals of empirical distribution functions and linear combinations of order statistics.