International Journal of Mathematics and Statistics Studies Vol.2, No.1, pp. 1-13, March 2014 Published by European Centre for Research Training and Development UK (www.ea-journals.org) 1 ON A SURVEY OF UNIFORM INTEGRABILITY OF SEQUENCES OF RANDOM VARIABLES S. A. Adeosun and S. O. Edeki 1 Department of Mathematical Sciences, Crescent University, Abeokuta, Nigeria. 2 Department of Industrial Mathematics, Covenant University, Cannanland, Ota, Nigeria. ABSTRACT: This paper presents explicitly a survey of uniformly integrable sequences of random variables. We also study extensively several cases and conditions required for uniform integrability, with the establishment of some new conditions needed for the generalization of the earlier results obtained by many scholars and researchers, noting the links between uniform integrability and pointwise convergence of a class of polynomial functions on conditional based. KEYWORDS: Uniform Integrability, Sequences, Boundedness, Convergence, Monotonicity. INTRODUCTION Uniform integrability is an important concept in functional analysis, real analysis, measure theory, probability theory, and plays a central role in the area of limit theorems in probability theory and martingale theory. Conditions of independence and identical distribution of random variables are basic in historic results due to Bernoulli, Borel and A.N. Kolmogorov[1]. Since then, serious attempts have been made to relax these strong conditions; for example, independence has been relaxed to pairwise independence. In order to relax the identical distribution condition, several other conditions have been considered, such as stochastic domination by an integrable random variable or uniform integrability in the case of weak law of large number. Landers and Rogge [2] prove that the uniform integrability condition is sufficient for a sequence of pairwise independence random variables in verifying the weak law of large numbers. Chandra [3] obtains the weak law of large numbers under a new condition which is weaker than uniform integrability: the condition of Ces ro uniform integrability. Cabrera [4], by studying the weak convergence for weighted sums of variables introduces the condition of uniform integrability concerning the weights, which is weaker than uniform integrability, and leads to Ces ro uniform integrabilty as a particular case. Under this condition, a weak law of large