SOME CHARACTERIZATIONS ON STATISTICAL

Let (Yn) be a sequence of random variables whose probabilitydistributions depend on x 2 [a; b]: It is well-known that ifnE(Yn x)2oconverges uniformly to zero on [a; b]; then, for all f 2 C[a; b]; fE (f(Yn))g isuniformly convergent to f on [a; b]; where E denotes the mathematical expecta-tion. In this paper, we mainly improve this result via the concept of statisticalconvergence from the summability theory, which is a weaker method than theusual convergence. Furthermore, we construct an example such that our newresult is applicable while the classical one is not.