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

[摘要] 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.

[学科分类] 数学（综合）

[关键词] stochastic sequences;martingale difference sequences;statistical convergence of martingale difference sequences;deferred weighted mean;Banach space;Korovkin-type theorem;Bernstein polynomial and positive linear operators