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Collection of research reports best universities honored author. Two. Le Van Dung, a number average convergence theorem for two index array of random elements in Banach spaces with integrable conditions are . Science (in Latin Scientia, meaning "knowledge "or" understanding ") is the efforts to implement the invention, and increased knowledge of the human understanding of how the operation of the physical world around them. Through controlled methods, scientists use to observe the signs of expression or of the material and unusual nature to collect data, analyze information to explain how it works, there at the phenomenon of things. One. | MEAN CONVERGENCE THEOREMS FOR DOUBLE ARRAYS OF RANDOM ELEMENTS IN BANACH SPACES UNDER SOME CONDITIONS OF UNIFORM INTEGRABILITY LE VAN DUNG a Abstract. In this paper we establish some mean convergence theorems for the double sums y U-ZX y v y Amnij Vij for an array of random elements Vij i j E Z in a real separable Banach space and an array of random variables Amnij xm i um yn j vn m 1 n 1 where xm m 1 um m 1 yn n 1 and vn n 1 are four sequences of integers. I. INTRODUCTION Consider an array Vij i j E Z of random elements defined on a probability space Q F P and an array Amnij xm i um yn j vn m 1 n 1 of random variables. Let xn n 1 un n 1 yn n 1 and vn n 1 be four sequences of integers such that un xn 0 for all n 1 and un xn TO as n TO vn yn 0 for all n 1 and vn yn TO as n TO. In this paper mean convergence theorems will be established. This convergence results are of the form um vn II EE Amnij Vij II 0. i . Limit theorems for weighted sums with or without random indies for random variables real-valued or Banach space-valued are studied by many authors in 1 4 7 . Recently M. Ordonez Cabrera and A. Volodin obtained a mean convergence theorem for the weighted sums J2V u Anj Vj in L1 under some conditions of uniform integrability see 2 . However the same problems for double sums have not yet been studied. II. PRELIMINARIES The expected value or mean of a random element V denoted by EV is defined to be the Pettis integral provided it exists. That is Vhas expected value EV E X if f EV Ef V for every f E X where X denotes the dual space of all continuous linear functionals on X. A sufficient condition for EV to exist is that EIIVII TO see e.g. Taylor 6 p.40 . 1 Nhận bài ngày 15 8 2007. sửa chữa xong ngày 12 12 2007. Consider an array of random variables Anj un j vn n 1 and an array Vj j E Z of random elements in a real separable Banach space X with a normal II II defined on a probability space Q F P . Le t Fn n 1 be a sequence of sub Ơ-algebras of F. For ea ch n 1 denote
