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• 1 Department of Statistics and Computing Science, Zheiang Gongshang University Hangzhou 310035, P. R. China Hangzhou 310035, P. R. China
• | 2 Department of Mathematics, Zhejiang University Hangzhou 310028, P. R. China Hangzhou 310028, P. R. China
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Summary

We provide uniform rates of convergence in the central limit theorem for linear negative quadrant dependent (LNQD) random variables. Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\{X_{n},\allowbreak n\ge1\}$ \end{document} be a LNQD sequence of random variables with \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $EX_{n}=0$ \end{document}, set \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $S_{n}=\sum_{j=1}^{n}X_{j}$ \end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $B_{n}^{2}=\text{Var}\, (S_{n})$ \end{document}. We show that \begin{gather*} \sup_{x} \left|P\left(\frac{S_{n}}{B_{n}}<x\right)-\Phi(x)\right|= O\bigg(n^{-\delta/(2+3\delta)}\vee \frac{n^{3\delta^{2}/(4+6\delta)}}{B^{2+\delta}_{n}} \sum_{i=1}^{n} E{|X_{i}|}^{2+\delta}\bigg) \end{gather*} under finite \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(2+\delta)$ \end{document}th moment and a power decay rate of covariances. Moreover, by the truncation method, we obtain a Berry--Esseen type estimate for negatively associated (NA) random variables with only finite second moment. As applications, we obtain another convergence rate result in the central limit theorem and precise asymptotics in the law of the iterated logarithm for NA sequences, and also for LNQD sequences.

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