Authors:
Jianfeng WangDepartment of Statistics and Computing Science, Zheiang Gongshang University Hangzhou 310035, P. R. China Hangzhou 310035, P. R. China

Search for other papers by Jianfeng Wang in
Current site
Google Scholar
PubMed
Close
and
Lixin ZhangDepartment of Mathematics, Zhejiang University Hangzhou 310028, P. R. China Hangzhou 310028, P. R. China

Search for other papers by Lixin Zhang in
Current site
Google Scholar
PubMed
Close
Restricted access

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.

  • Collapse
  • Expand

To see the editorial board, please visit the website of Springer Nature.

Manuscript Submission: HERE

For subscription options, please visit the website of Springer Nature.

Acta Mathematica Hungarica
Language English
Size B5
Year of
Foundation
1950
Volumes
per Year
3
Issues
per Year
6
Founder Magyar Tudományos Akadémia  
Founder's
Address
H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
Springer Nature Switzerland AG
Publisher's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher
Chief Executive Officer, Akadémiai Kiadó
ISSN 0236-5294 (Print)
ISSN 1588-2632 (Online)

Monthly Content Usage

Abstract Views Full Text Views PDF Downloads
Aug 2022 0 0 0
Sep 2022 0 0 0
Oct 2022 2 0 0
Nov 2022 0 0 0
Dec 2022 1 0 0
Jan 2023 1 0 0
Feb 2023 0 0 0