SONGKLANAKARIN JOURNAL OF SCIENCE & TECHNOLOGY

Volume 41, No. 03, Month MAY, Year 2019, Pages 567 - 579

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

Nagarajan Subramanian, Ayhan Esi, Mustafa Kemal Ozdemir

Abstract Download PDF

This paper aims to improve further on the work of Phu (2001), Aytar (2008), and Ghosal (2013). We propose a new apporach to extend the application area of rough statistical convergence usually used in triple sequence of the Bernstein operator of real numbers to the theory of probability distributions. The introduction of this concept in the probability of Bernstein polynomials of rough statistical convergence, Bernstein polynomials of rough strong Cesàro summable, Bernstein polynomials of rough lacunary statistical convergence, Bernstein polynomials of rough convergence, Bernstein polynomials of rough statistical convergence, and Bernstein polynomials of rough strong summable to generalize the convergence analysis to accommodate any form of distribution of random variables. Among these six concepts in probability only three convergences are distinct Bernstein polynomials of rough statistical convergence: (1) Bernstein polynomials of rough lacunary statistical convergence, (2) Bernstein polynomials of rough statistical convergence where Bernstein polynomials of rough strong Cesàro summable is equivalent to Bernstein polynomials of rough statistical convergence, and (3) Bernstein polynomials of rough convergence which is equivalent to Bernstein polynomials of rough lacunary statistical convergence. Bernstein polynomials 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 in these classes and in this way we demonstrated that rough statistical convergence in probability is the more generalized concept than the usual Bernstein polynomials of rough statistical convergence.

Keywords

rough statistical convergence, rough strong Ces?ro summable, rough lacunary statistical convergence, rough convergence, rough ?- statistical convergence, rough strong summable, Bernstein polynomials

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