Rough statistical convergence on triple sequence of the Bernstein operator of random variables in probability

摘要

This paper aims to improve further on the work of Phu (2001), Aytar (2008), and Ghosal (2013). We propose a newapporach to extend the application area of rough statistical convergence usually used in triple sequence of the Bernstein operatorof real numbers to the theory of probability distributions. The introduction of this concept in the probability of Bernsteinpolynomials of rough statistical convergence, Bernstein polynomials of rough strong Cesàro summable, Bernstein polynomials ofrough lacunary statistical convergence, Bernstein polynomials of rough convergence, Bernstein polynomials of roughstatistical convergence, and Bernstein polynomials of rough strong summable to generalize the convergence analysis toaccommodate any form of distribution of random variables. Among these six concepts in probability only three convergences aredistinct Bernstein polynomials of rough statistical convergence: (1) Bernstein polynomials of rough lacunary statisticalconvergence, (2) Bernstein polynomials of rough statistical convergence where Bernstein polynomials of rough strongCesàro summable is equivalent to Bernstein polynomials of rough statistical convergence, and (3) Bernstein polynomials ofrough convergence which is equivalent to Bernstein polynomials of rough lacunary statistical convergence. Bernsteinpolynomials of rough strong summable is equivalent to Bernstein polynomials of rough statistical convergence.Basic properties and interrelations of these three distinct convergences are investigated and some observations were made inthese classes and in this way we demonstrated that rough statistical convergence in probability is the more generalized conceptthan the usual Bernstein polynomials of rough statistical convergence.