Korovkin-Type Approximation Theorems for Positive Linear Operators via Statistical Martingale Sequences

Résumé

In this paper, we introduce and study the notions of statistical product convergence and statistical product summability via deferred Ces\`{a}ro and deferred N\"{o}rlund product means for martingale sequences of random variables. After that, we establish an inclusion theorem concerning the relation between
these two lovely and potentially useful notions. We also describe and prove a set of new Korovkin-type approximation theorems for a martingale sequence over a Banach space based on the principles we have put forward. Additionally, we demonstrate that most (if not all) of the prior findings both in statistical and classical forms can be improved upon by our approximation theorems. Finally, we offer an example of martingale sequence by using the generalized Bernstein polynomials to show that our established theorems are significantly stronger than the traditional and statistical versions of several theorems already exist in the literature.