Statistical convergence of martingale difference sequence via deferred weighted mean and Korovkin-type theorems

Abstract

In the present paper, we introduce and study the concepts of statistical convergence and statistical summability for martingale difference sequences of random variables via deferred weighted summability mean. We then establish an inclusion theorem concerning the relation between these two beautiful concepts. Also, based upon our proposed notions, we state and prove new Korovkin-type approximation theorems with algebraic test functions for a martingale difference sequence over a Banach space and demonstrate that our theorems effectively extend and improves most (if not all) of the previously existing results (in statistical and classical versions). Finally, we present an illustrative example by using the generalized Bernstein polynomial of a martingale difference sequence in order to demonstrate that our established theorems are stronger than its traditional and statistical versions.

A new approach to statistical Riemann-Stieltjes integrals

“…In particular, it is very much useful in the study of Machine Learning, Soft Computing, Number Theory, Measure theory, Probability theory, and so on. For some recent research works in this direction, see [1], [4], [5], [6], [7], [8], [9], [12], [14], [19] and [21].…”

“…Subsequently, Srivastava et al [18] established another result on statistical product convergence of martingale sequences and its applications to Korovkin-type approximation theorems. Recently, Jena and Paikray [6] demonstrated statistical convergence of martingale difference sequence via deferred weighted mean and proved a Korovkin-type theorem. Moreover, such concepts have been generalized in many aspects.…”