Product of statistical probability convergence and its applications to Korovkin-type theorem

Abstract

In the present work, we introduce and study the notion of statistical probability convergence for sequences of random variables as well as the concept of statistical convergence for sequences of real numbers, which are defined over a Banach space via product of deferred Cesàro and deferred Nörlund summability means. We first establish a theorem presenting a connection between them. Based upon our proposed method, we then prove a Korovkin-type approximation theorem with algebraic test functions for a sequence of random variables on a Banach space and demonstrate that our theorem effectively extends and improves most (if not all) of the previously existing results (in classical as well as statistical versions). Finally, an illustrative example is presented here by means of the generalized Meyer-König and Zeller operators for a positive sequence of random variables in order to demonstrate that our established theorem is stronger than its traditional and statistical versions.

Strong Riesz summability of Fourier series

“…Also, the statistical convergene and statistical summability are more general than the ordinary convergene and ordinary summability. For recent works in this direction, see [1], [5], [6], [8], [9], [10], [11], [15], [16], [17], [18], [19] and [20]. Dealing with strong summability of Nörlund mean Mittal [13] has proved a theorem as follows.…”

Generalized Nörlund and Nörlund-type Means of Sequences of Fuzzy Numbers

“…Also, Das et al [2] used statistical (C, 1)(E, µ) product summability mean for sequences of fuzzy numbers to prove a fuzzy Korovkin-type approximation theorem. For more studies in this direction one may refer to [3], [4], [5], [7], [8], [14], [15] and [16]. Motivated essentially by the above mentioned works, we investigate here the characterization of generalized Nörlund and Nörlund-type (Riesz) means of sequences of fuzzy real numbers.…”

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

“…Moreover, the introduction of statistical probability convergence has enhanced the glory of this development. For some recent research works in this direction, see previous studies 3–15 …”