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References [1] J.-N . Bacro and M . Brito . On Mason’s extension of the Erdős–Rényi law of large numbers . Statist. Probab. Lett ., 11 : 43 – 47 , 1991 . [2] J.-N . Bacro , P . Deheuvels , and J . Steinebach . Exact convergence rates in

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. 48 600 607 ASMUSSEN, S. and KURTZ, T., Necessary and sufficient conditions for complete convergence in the law of large numbers, Ann. Probab. 8

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Abstract  

Marcinkiewicz laws of large numbers for φ-mixing strictly stationary sequences with r-th moment barely divergent, 0 < r < 2, are established. For this dependent analogs of the Lévy-Ottaviani-Etemadi and Hoffmann-Jørgensen inequalities are revisited.

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In a one-parameter model for evolution of random trees strong law of large numbers and central limit theorem are proved for the number of vertices with low degree. The proof is based on elementary martingale theory.

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Abstract  

An exponential inequality for the tail of the conditional expectation of sums of centered independent random variables is obtained. This inequality is applied to prove analogues of the Law of the Iterated Logarithm and the Strong Law of Large Numbers for conditional expectations. As corollaries we obtain certain strong theorems for the generalized allocation scheme and for the nonuniformly distributed allocation scheme.

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Franck, W. E. and Hanson, D. L. , Some results giving rates of convergence in the law of large numbers for weighted sums of independent variables,. Trans. Amer. Math. Soc. , 124 (1966), 347–359. MR 33 #8017

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.1007/BF01895656 . [4] Móricz , F. 1976 Moment inequalities and the strong laws of large numbers Z. Wahrscheinlichkeitstheorie und verw

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Révész, P. , The Laws of Large Numbers , Akadémiai Kiadó, Budapest, 1967. Révész P. The Laws of Large Numbers 1967

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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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1363 1386 ERDÖS, P. and RÉNYI, A., On a new law of large numbers, J. Analyse. Math . 23 (1970), 103-111. MR 42 #6907 On a new law of large numbers

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