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 Nörlund 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-König 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.

中文翻译：

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通过递延 Nörlund 均值的统计概率收敛及其在近似定理中的应用

递延加权统计概率收敛的概念最近引起了研究人员的广泛关注，主要是因为它比递延加权统计收敛更通用。这些概念是由 Srivastava 等人引入和研究的。（Appl 肛门离散数学，2020 年）。在目前的工作中，我们介绍并研究了统计概率收敛的概念，以及通过递延 Nörlund 可和均值分别定义在 Banach 空间上的随机变量序列和实数序列的统计收敛性。我们还建立了一个定理，展示了这两个有趣概念之间的联系。此外，基于我们提出的方法，我们已经证明了一个新的 Korovkin 型近似定理，它带有代数测试函数，用于 Banach 空间上的一系列随机变量，并证明我们的定理有效地扩展和改进了大部分（如果不是全部）先前存在的结果（在统计版本中）。最后，这里通过随机变量序列的广义 Meyer-König 和 Zeller 算子提供了一个说明性示例，以证明我们建立的定理比其传统和统计版本更强。