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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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