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: Boundedness, Convergence, Monotonicity, Sequences, Uniform Integrability
