Statistical probability convergence via the deferred Nörlund mean and its applications to approximation theorems

Abstract

The notion of deferred weighted statistical probability convergence has recently attracted the wide-spread attention of researchers due mainly to the fact that it is more general than the deferred weighted statistical convergence. Such concepts were introduced and studied by Srivastava et al. (Appl Anal Discrete Math, 2020). In the present work, we introduced and studied the notion of statistical probability convergence as well as statistical convergence for sequences of random variables and sequences of real numbers respectively defined over a Banach space via deferred Norlund summability mean. We have also established a theorem presenting a connection between these two interesting notions. Moreover, based upon our proposed methods, we have proved a new Korovkin-type approximation theorem with algebraic test functions for a sequence of random variables on a Banach space and demonstrated that our theorem effectively extends and improves most (if not all) of the previously existing results (in statistical versions). Finally, an illustrative example is presented here by the generalized Meyer-Konig and Zeller operators of a sequence of random variables in order to demonstrate that our established theorem is stronger than its traditional and statistical versions.