báo cáo hóa học: Some exponential inequalities for acceptable random variables and complete convergence

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí sinh học đề tài : Some exponential inequalities for acceptable random variables and complete convergence
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báo cáo hóa học:"Some exponential inequalities for acceptable random variables and complete convergence"Journal of Inequalities andApplications This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Some exponential inequalities for acceptable random variables and complete convergence Journal of Inequalities and Applications 2011, 2011:142 doi:10.1186/1029-242X-2011-142 Aiting Shen (shenaiting1114@126.com) Shuhe Hu (hushuhe@263.net) Andrei Volodin (volodin@math.uregina.ca) Xuejun Wang (wxjahdx2000@126.com) ISSN 1029-242X Article type Research Submission date 6 July 2011 Acceptance date 22 December 2011 Publication date 22 December 2011 Article URL http://www.journalofinequalitiesandapplications.com/content/2011/1/ This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Journal of Inequalities and Applications go to http://www.journalofinequalitiesandapplications.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2011 Shen et al. ; licensee Springer.This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Some exponential inequalities for acceptablerandom variables and complete convergence Aiting Shen1 , Shuhe Hu1 , Andrei Volodin∗2 and Xuejun Wang1 1 School of Mathematical Science, Anhui University, Hefei 230039, China2 Department of Mathematics and Statistics, University of Regina, Regina Saskatchewan S4S 0A2, Canada ∗ Corresponding author: volodin@math.uregina.ca Email addresses: AS: baret@sohu.com SH: hushuhe@263.net XW: wxjahdx2000@126.com December 14, 2011 Abstract 1 Some exponential inequalities for a sequence of acceptable random variables are obtained, such as Bernstein-type inequality, Hoeffding-type inequality. The Bernstein-type inequality for acceptable random variables generalizes and improves the corresponding results presented by Yang for NA random variables and Wang et al. for NOD random variables. Using the exponential inequalities, we further study the complete convergence for acceptable random variables. MSC(2000): 60E15, 60F15. Keywords: acceptable random variables; exponential inequality; complete conver- gence.1 IntroductionLet {Xn , n ≥ 1} be a sequence of random variables defined on a fixed probability space n(Ω, F , P ). The exponential inequality for the partial sums i=1 (Xi − EXi ) plays animportant role in various proofs of limit theorems. In particular, it provides a measureof convergence rate for the strong law of large numbers. There exist several versionsavailable in the literature for independent random variables with assumptions of uniformboundedness or some, quite relaxed, control on their moments. If the independent case isclassical in the literature, the treatment of dependent variables is more recent. First, we will recall the definitions of some dependence structure.Definition 1.1. A finite collection of random variables X1 , X2 , . . . , Xn is said to benegatively associated (NA) if for every pair of disjoint subsets A1 , A2 of {1, 2, . . . , n}, Cov {f (Xi : i ∈ A1 ), g (Xj : j ∈ A2 )} ≤ 0, (1.1)whenever f and g are coordinatewise nondecreasing (or coordinatewise nonincreasing ...