Weighted statistical convergence of order α and its applications

Abstract

The definition of weighted statistical convergence was first introduced by Karakaya and Chishti (2009) [1]. After that the definition was modified by Mursaleen et al. (2012) [2]. But some problems are still there; so it will be further modified in this paper. Using it some newly developed definitions of the convergence of a sequence of random variables in probability have been introduced and their interrelations also have been investigated, and in this way a partial answer to an open problem posed by Das and Savas (2014) [3] has been given.

A New Type of Generalization on W—Asymptotically J λ—Statistical Equivalence with the Number of α

“…The new definition is not exactly parallel to that of statistical convergence. Some other applications of this concept are the λ−statistical convergence of order α by Çolak and Bektaş [24], the lacunary statistical convergence of order α byŞengül and Et [25], the weighted statistical convergence of order α and its applications by Ghosal [26], and the almost statistical convergence of order α by Et, Altın and Çolak [27]. J −statistical convergence and J −lacunary statistical convergence of order α were introduced by Das and Savaş in 2014 [28].…”

Weighted βγ-summability of fuzzy functions of order θ

“…Definition 2.4 (Weighted strong Cesaro summability). [7] Let (t n ) be a sequence of nonnegative real numbers such that t 1 > 0,…”

“…β([λγ])(n) T βγ(n) − T β([λγ])(n) Ŝβ([λγ])(n) (x) x ∈ [a, b] T β([λγ])(n) T βγ(n) − T β([λγ])(n) Ŝβ([λγ])(n) (x) + f (x) − 2ε 3 T β([λγ])(n) T βγ(n) − T β([λγ])(n) Ŝβγ(n) (x) + Ŝβγ(n) (x)(from[7] and[11])= T β([λγ])(n) T βγ(n) − T β([λγ])(n) Ŝβ([λγ])(n) (x) + 1 T βγ(n) − T β([λγ])(n) k∈ [λγn]+1,γn t k fk (x) (follows from [10]) T β([λγ])(n) T βγ(n) − T β([λγ])(n) Ŝβ([λγ])(n) (x) + fλn (x) + ε 3 since fn (x) is slowly decreasing. Thus f (x) − ε fγn (x)(12)Combining[9] and[12], we getf (x) − ε fγn (x) f (x) + ε, x ∈ [a, b]So fγ k (x) converges to f (x) for all x ∈ [a, b].…”

Generalized weighted statistical convergence of double sequences and applications

“…Recently Ghosal [17] was added to the definition of weighted statistical convergence the condition lim inf p n > 0.…”