Equi-Statistical Convergence of a Sequence of Distribution Functions via Deferred N\"{o}rlund Summability Mean and Associated Approximation Theorems

Abstract

In this paper, the notion of statistical point-wise convergence,
equi-statistical convergence and statistical uniform convergence
of a sequence of distribution functions via the deferred
N\"{o}rlund summability mean has been introduced, and accordingly
an inclusion relation between these interesting notions is
established. Moreover, as an application point of view, a new
Korovkin-type approximation theorem is proved via the deferred
N\"{o}rlund equi-statistical convergence for the sequence of
distribution functions. Also, some illustrative examples are
considered to justify that the proposed theorem is a nontrivial
extension of some well established Korovkin-type approximation
theorems for sequence of real-valued functions. Finally, a number
of interesting cases are highlighted in support of the definitions
and outcomes.