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Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Some exponential inequalities for acceptable random variables and complete convergence | Shen et al. Journal of Inequalities and Applications 2011 2011 142 http www.journalofinequalitiesandapplications.eom content 2011 1 142 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Some exponential inequalities for acceptable random variables and complete convergence Aiting Shen 1 Shuhe Hu1 Andrei Volodin2 and Xuejun Wang1 Correspondence volodin@math. uregina.ca department of Mathematics and Statistics University of Regina Regina Saskatchewan S4S 0A2 Canada Full list of author information is available at the end of the article Springer Abstract 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 convergence 1 Introduction Let Xn n 1 be a sequence of random variables defined on a fixed probability space En I Xi EXi plays an important role in various proofs of limit theorems. In particular it provides a measure of convergence rate for the strong law of large numbers. There exist several versions available in the literature for independent random variables with assumptions of uniform boundedness or some quite relaxed control on their moments. If the independent case is classical 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 be negatively associated NA if for every pair of disjoint subsets A1 A2 of 1 2 . n Cov f Xi i e A1 g Xj j e A2 l 0 1.1 whenever f and g are coordinatewise
