Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : A note on the complete convergence for arrays of dependent random variables | Sung Journal of Inequalities and Applications 2011 2011 76 http content 2011 1 76 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access A note on the complete convergence for arrays of dependent random variables Soo Hak Sung Correspondence sungsh@ Department of Applied Mathematics Pai Chai University Taejon 302-735 South Korea Abstract A complete convergence result for an array of rowwise independent mean zero random variables was established by Kruglov et al. 2006 . This result was partially extended to negatively associated and negatively dependent mean zero random variables by Chen et al. 2007 and Dehua et al. 2011 respectively. In this paper we obtain complete extended versions of Kruglov et al. Mathematics Subject Classification 60F15 Keywords Complete convergence Negatively associated random variables Negatively dependent random variables 1 Introduction The concept of complete convergence was introduced by Hsu and Robbins 1 . A sequence Xn n 1 of random variables is said to converge completely to the constant 0 if ỄP Xn - el c ro for all o 0. n 1 Hsu and Robbins 1 proved that the sequence of arithmetic means of . random variables converges completely to the expected value if the variance of the summands is finite. Erdos 2 proved the converse. The result of Hsu-Robbins-Erdos has been generalized and extended in several directions by many authors. Sung et al. 3 see also Hu et al. 4 obtained the following complete convergence theorem for arrays of rowwise independent random variables Xni 1 i kn n 1 where kn n 1 is a sequence of positive integers. Theorem . Let Xni 1 i kn n 1 be an array of rowwise independent random variables and an n 1 a sequence of nonnegative constants. Suppose that the following conditions hold i ED1 a M P l A n l e ro for aU e 0 ii there exist J 2 and s 0 such that s o CO i kn a EX2j Xml ro Springer 2011 Hak Sung licensee Springer. This is .
