Some Characterizations on Statistical Convergence of Expected Values of Random Variables

Abstract

Let (Y-n) be a sequence of random variables whose probability distributions depend on x is an element of [a, b]. It is well-known that if {E (Y-n - x)(2)} converges uniformly to zero on [a, b], then, for all f is an element of C[a, b], {E (f (Y-n))1 is uniformly convergent to f on [a, b], where E denotes the mathematical expectation. In this paper, we mainly improve this result via the concept of statistical convergence from the summability theory, which is a weaker method than the usual convergence. Furthermore, we construct an example such that our new result is applicable while the classical one is not.