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CHATTERJI, S. D., A principle of subsequences in probability theory: The central limit theorem, Adv. Math. 13 (1974), 31-54; correction, ibid. 14 (1974), 266-269. MR 49 #6312 A principle of subsequences in probability theory

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2010 PAP, G., Central limit theorems on nilpotent Lie groups, Probab. Math. Statist. 14 (1993), 287-312. MR 96c :22010 Central limit theorems on nilpotent Lie

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Bolthausen, E. , On a functional central limit theorem for random walks conditioned to stay positive, The Annals of Probability , 4 , No. 3, 480–485, 1976. Bolthausen E. On a functional

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Hamana, Y. , On the central limit theorem for the multiple point range of random walk, J. Fac. Sci. Univ. Tokyo 39 (1992), 339–363. MR 93h :60112 Hamana Y. On the central

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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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Abstract  

Let X 1,X 2,... be a sequence of independent and identically distributed random variables, and put
\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$S_n = X_1 + \cdot \cdot \cdot + X_n$$ \end{document}
. Under some conditions on the positive sequence τ n and the positive increasing sequence a n, we give necessary and sufficient conditions for the convergence of
\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sum\nolimits_{n = 1}^\infty {\tau _n } P\left( {\left| {S_n } \right| \geqslant \varepsilon an} \right)$$ \end{document}
for all & > 0, generalizing Baum and Katz's~(1965) generalization of the Hsu–Robbins–Erds (1947, 1949) law of large numbers, also allowing us to characterize the convergence of the above series in the case where τn = n -1and
\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$an = \left( {n\log n} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}$$ \end{document}
for n ≤ 2, thereby answering a question of Spătaru. Moreover, some results for non-identically distributed independent random variables are obtained by a recent comparison inequality. Our basic method is to use a central limit theorem estimate of Nagaev (1965) combined with the Hoffman-Jrgensen inequality~(1974).
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Fukuyama, K. , The central limit theorem for Riesz-Raikov sums, Prob. Theory Related Fields , 100 (1994), no. 1, 57–75. MR 1292190 ( 95i :60020) Fukuyama K. The central limit theorem

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Zhang, L. X. , A functional central limit theorem for asymptotically negatively dependent random fields, Acta Math. Hungar. , 86 (3) (2000a), 237–259. MR 1756175 ( 2001c :60055) Zhang L. X

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589 Bradley, R. C. , A central limit theorem for stationary ρ -mixing sequences with infinite variance, Ann. Probab. , 16 (1988), 313–332. MR 89a :60053

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] Hall , P. , Heyde , C. C. 1980 Martingale Limit Theory and Its Applications Academic Press New York . [6] Johnson , O. , Samworth , R. 2005 Central limit theorem and convergence to

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