Statistical product convergence of martingale sequences and its applications to Korovkin-type approximation theorems

摘要

In this paper, we investigate and study the concepts of statistical product convergence and statistical product summability via deferred Cesaro and deferred weighted product means for martingale sequences of random variables. We then establish an inclusion theorem concerning the relation between these two nice and potentially useful concepts. Also, based upon our proposed concepts, we state and prove a set of new Korovkin-type approximation theorems for a martingale sequence over a Banach space. Moreover, we demonstrate that our approximation theorems effectively extend and improve most (if not all) of the previously existing results (both in statistical and classical versions). Finally, by using the generalized Bernstein polynomials, we present an illustrative example of a martingale sequence in order to demonstrate that our established theorems are quite stronger than the traditional and statistical versions of different theorems existing in the literature. We also suggest a direction for future researches on this subject, which are based upon the basic (or q-) calculus, but not upon the trivial and inconsequential variations involving the so-called (p, q)-calculus.