Improved results in almost sure central limit theorems for the maxima and partial sums for Gaussian sequences

被引:0
|
作者
Qunying Wu
机构
[1] Guilin University of Technology,College of Science
来源
Journal of Inequalities and Applications | / 2015卷
关键词
standardized Gaussian sequence; maxima; partial sums and maxima; almost sure central limit theorem; 60F15;
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摘要
Let X,X1,X2,…\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$X, X_{1}, X_{2},\ldots$\end{document} be a standardized Gaussian sequence. The universal results in almost sure central limit theorems for the maxima Mn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_{n}$\end{document} and partial sums and maxima (Sn/σn,Mn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(S_{n}/\sigma_{n}, M_{n})$\end{document} are established, respectively, where Sn=∑i=1nXi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S_{n}=\sum_{i=1}^{n}X_{i}$\end{document}, σn2=VarSn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sigma^{2}_{n}=\operatorname{Var}S_{n}$\end{document}, and Mn=max1≤i≤nXi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_{n}=\max_{1\leq i\leq n}X_{i}$\end{document}.
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