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# Some multivariate inequalities with applications

Periodica Mathematica Hungarica
Authors: Sana Louhichi and Sofyen Louhichi

## Summary

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} ${\cal {X}}_{n} =(X_1,\ldots,X_n)$ \end{document} be a random vector. Suppose that the random variables \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} $(X_i)_{1\leq i\leq n}$ \end{document} are stationary and fulfill a suitable dependence criterion. 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} $f$ \end{document} be a real valued function defined on \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} $\mathbbm{R}^n$ \end{document} having some regular properties. 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} ${\cal {Y}}_{n}$ \end{document} be a random vector, independent 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} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} ${\cal {X}}_{n}$ \end{document}, having independent and identically distributed components. We control \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} $\left|\mathbbm{E}(f({\cal {X}}_{n}))-\mathbbm{E} (f({\cal {Y}}_{n}))\right|$ \end{document}. Suitable choices of the function \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} $f$ \end{document} yield, under minimal conditions, to rates of convergence in the central limit theorem, to some moment inequalities or to bounds useful for Poisson approximation. The proofs are derived from multivariate extensions of Taylor's formula and of the Lindeberg decomposition. In the univariate case and in the mixing setting the method is due to Rio (1995).

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# A characterization of signed discrete infinitely divisible distributions

Studia Scientiarum Mathematicarum Hungarica
Authors: Huiming Zhang, Bo Li, and G. Jay Kerns

Mathematicum , 10 ( 6 ) ( 1998 ), 687 – 698 .  Baez-Duarte , L. , Central limit theorem for complex measures , Journal of

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# A note on sub-independent random variables and a class of bivariate mixtures

Studia Scientiarum Mathematicarum Hungarica
Authors: G. Hamedani, Hans Volkmer, and J. Behboodian

central limit theorem, Arch. Math. , 43 (1984), no. 3, pp. 258–264. MR 0766433 ( 86a :60029) Walter G. G. A fixed point theorem and its application to the central limit theorem

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# On the strong law of large numbers for ϕ-mixing and ρ-mixing random variables

Acta Mathematica Hungarica
Author: Anna Kuczmaszewska

.  Peligrad , M. 1987 On the central limit theorem for ρ -mixing sequences of random variables Ann. Probab. 15 1387 – 1394 10.1214/aop

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# Bounds on moments of symmetric statistics

Studia Scientiarum Mathematicarum Hungarica
Authors: R. Ibragimov and Sh. Sharakhmetov

. 22 1044 1077 EGOROV, V. A., On a central limit theorem with random normalization, Rings and modules. Limit theorems of probability theory , No. 1

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# Small ball estimates for Brownian motion under a weighted sup-norm

Studia Scientiarum Mathematicarum Hungarica
Authors: Ph. Berthet and Z. Shi

. 8 361 386 ACOSTA, A. DE, Small deviations in the functional central limit theorem with applications to functional laws of the iterated logarithm

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# A rejoinder on energy versus impact indicators

Scientometrics
Authors: Loet Leydesdorff and Tobias Opthof

and uses averages on the assumption of the Central Limit Theorem (Glänzel 2010 ). However, citation distributions are extremely skewed (Seglen 1992 , 1997 ; cf. Leydesdorff 2008 ) and central tendency statistics give misleading results. Using

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# Teachers and prospective teachers’ conceptions about averages

Journal of Adult Learning, Knowledge and Innovation
Authors: Karin Landtblom and Lovisa Sumpter

. , & Clark , J. ( 2003 ). Successful students’ conceptions of mean, standard deviation, and the Central Limit Theorem (Unpublished paper). Retrieved from http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.446.6077&rep=rep1&type

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# Universality of performance indicators based on citation and reference counts

Scientometrics
Authors: T. S. Evans, B. S. Kaube, and N. Hopkins

that its variance changes with time. For the case β = 0 the central limit theorem tells us that the variance should scale as σ 2 ∼ t −1 where t denotes the number of elapsed time steps. This would be manifested in a systematic temporal variation in

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