A new version of uniform integrability via power series summability methods

Uniform integrability is an interesting concept in probability theory and functional analysis since it plays an important role in establishing laws of large numbers. In the literature, there are several versions of uniform integrability. Some are defined with the help of matrix summability methods, such as the Cesàro matrix, or more general methods. In this paper, we introduce a new version of uniform integrability via power series summability methods. We investigate the relationships of this new concept with some previous concepts and give $L_1$- and $L_2$-convergence results for the laws of large numbers.