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    Generalized Wijsman rough Weierstrass statistical six dimensional triple geometric difference sequence spaces of fractional order defined by Musielak-Orlicz function of interval numbers
    (Balkan Society of Geometers, 2019) Subramanian N.; Esi A.; Ozdemir M.K.
    We generalized the concepts in probability of Wijsman rough lacunary statistical by introducing the interval numbers of Weierstrass of fractional order, where ? is a proper fraction and ? = (?mnk) is any fixed sequence of nonzero real or complex numbers. We study some properties of this operator involving Wijsman rough lacunary sequence ? of interval numbers and arbitrary sequence p = (prst) of strictly positive real numbers and investigate the topological structures of related six dimensional triple geometric difference sequence spaces of interval numbers. In this study, we consider a generalization for Weierstrass rough six dimensional triple geometric difference sequence of these metric spaces by taking a ? function, satisfying the following conditions. Let ?m,n,k be a positive function for all m, n, k ? N such that (i) lim m,n,k?? ?mnk = 0, (ii) ?3?mnk = ?mnk - ?m,n+1,k - ?m,n,k+1 + ?m,n+1,k+1 - ?m+1,n,k + ?m+1,n+1,k + ?m+1,n,k+1 - ?m+1,n+1,k+1 ? 0: or ?mnk = 1. Therefore, according to class of functions which satisfying the conditions (i) and (ii) with metric spaces of six dimensional triple geometric difference sequence spaces of interval numbers defined by a Musielak-Orlicz function. © Balkan Society of Geometers, Geometry Balkan Press 2019.
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    Rough statistical convergence on triple sequence of the Bernstein operator of random variables in probability
    (Prince of Songkla University, 2019) Subramanian N.; Esi A.; Ozdemir M.K.
    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 N? convergence, Bernstein polynomials of rough statistical convergence, and Bernstein polynomials of rough strong (V, ?)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 N?-convergence which is equivalent to Bernstein polynomials of rough lacunary statistical convergence. Bernstein polynomials of rough strong (V, ?)-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. © 2019, Prince of Songkla University. All rights reserved.