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 Summability of Double Independent Random Variables | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 948195 12 pages doi 2008 948195 Research Article Summability of Double Independent Random Variables Richard F. Patterson1 and Ekrem Sava 2 1 Department of Mathematics and Statistics University of North Florida 1 UNF Drive Jacksonville FL 32224 uSa 2 Department of Mathematics Istanbul commerce University Uskudar 34672 Istanbul Turkey Correspondence should be addressed to Richard F. Patterson rpatters@ Received 21 May 2008 Accepted 1 July 2008 Recommended by Jewgeni Dshalalow We will examine double sequence to double sequence transformation of independent identically distribution random variables with respect to four-dimensional summability matrix methods. The main goal of this paper is the presentation of the following theorem. If maxt am n k i maxkfa ka i OFm 1 O n-Y2 Y1 Y2 0 then E X 1 1 Y1 OT and E X 1 1 Y2 OT imply that Ymn y almost sure P-convergence. Copyright 2008 R. F. Patterson and E. Savas. 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. 1. Introduction Let Xw be a factorable double sequence of independent identically distributed random variables with E Xk i OT and E Xk t y. Let A am n k l be a factorable double sequence to double sequence transformation defined as OT OT .Ax m n V am n k lxk l. k l 1 1 These factorable sequences and matrices will be used to characterize such transformations with respect to Robison and Hamilton-type conditions see 1 2 . That is regularity conditions of the following type. The four-dimensional matrix A is RH-regular if and only if RH1 P-limm nam n k l 0 for each k and l RH2 P-limm n k lam n k l 1 RH3 P-limm nEk Om k i 0 for each l RH4 P-limm nEi a M1 0 for each k RH5 Zkj am n k i is P-convergent and RH6 there exist positive numbers A and B such that Xk
