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Báo cáo hóa học: " Research Article A Hajek-Renyi-Type Maximal Inequality and ´ ´ Strong Laws of Large Numbers for "

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article A Hajek-Renyi-Type Maximal Inequality and ´ ´ Strong Laws of Large Numbers for | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 569759 14 pages doi 10.1155 2010 569759 Research Article A Hajek-Renyi-Type Maximal Inequality and Strong Laws of Large Numbers for Multidimensional Arrays Nguyen Van Quang1 and Nguyen Van Huan2 1 Department of Mathematics Vinh University Nghe An 42000 Vietnam 2 Department of Mathematics Dong Thap University Dong Thap 871000 Vietnam Correspondence should be addressed to Nguyen Van Huan vanhuandhdt@yahoo.com Received 1 July 2010 Accepted 27 October 2010 Academic Editor Alexander I. Domoshnitsky Copyright 2010 N. V. Quang and N. Van Huan. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. A Hajek-Renyi-type maximal inequality is established for multidimensional arrays of random elements. Using this result we establish some strong laws of large numbers for multidimensional arrays. We also provide some characterizations of Banach spaces. 1. Introduction and Preliminaries Throughout this paper the symbol C will denote a generic positive constant which is not necessarily the same one in each appearance. Let d be a positive integer the set of all nonnegative integer d-dimensional lattice points will be denoted by Nd and the set of all positive integer d-dimensional lattice points will be denoted by Nd. We will write 1 m n and n 1 for points 1 1 . 1 m1 m2 . . md n1 n2 . . nà and n1 1 n2 1 . nd 1 respectively The notation m n or n m means that mi n for all i 1 2 . d the limit n TO is interpreted as ni TO for all i 1 2 . d this limit is equivalent to min n1 n2 . . nd to and we define n nd 1ni. Let bn n e Nd be a d-dimensional array of real numbers. We define Abn to be the dth-order finite difference of the b s at the point n. Thus bn 1 k n Abk for all n e Nd. For example if d 2 then for all i j e N2 Abij bij - bi j-1 - bi-1j