Restricting statistical convergence

Abstract

We first introduce a new notion called statistical convergence of order α and primarily show that it gives rise to a decreasing chain of closed linear subspaces of the space of all bounded real sequences with sup norm which never coincides with the class of convergent sequences and in fact their intersection properly contains the class of convergent sequences. We then show that the same method can be applied for double sequences also and introduce the notion of statistical convergence of order (α, β).

New approach to the concept of aᶥ -statistical convergence with the number of alpha

“…It should be mentioned that this concept has not been studied until now and our results e¤ectively extend and improve all the existing results in [24] ; [4], [2] ; [7].…”

“…In [2] and [4] ; a di¤erent direction was given to the statistical convergence, where the notion of statistical convergence of order (0 < < 1) was introduced by using the notion of natural density of order : The behavior of this new convergence was not exactly parallel to that of statistical convergence. Some other applications of this concept are statistical convergence of order by Çolak and Bektaş ( [5]) ; lacunary statistical convergence of order byŞengül and Et ( [23]), weighted statistical convergence of order and its applications by Ghosal ([13]) and almost statistical convergence of order by Et, Alt¬n and Çolak ( [9]).…”

<b>Almost Statistical Convergence of Order alpha

“…Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, Ergodic theory, Number theory, Measure theory, Trigonometric series, Turnpike theory and Banach spaces. Later on it was further investigated from the sequence space point of view and linked with summability theory by Šalát (1980), Fridy (1985), Connor (1988), Rath and Tripathy (1994), Savaş (2000), Mursaleen (2000), Miller and Orhan (2001), Et and Nuray (2001), Mursaleen et al (2001Mursaleen et al ( , 2003Mursaleen et al ( , 2009), Mursaleen and Edely (2009), Lohani (2009), Çolak (2010), Çolak and Bektaş (2011), Bhunia et al (2012), Kumar and Mursaleen (2011) and Savaş and Mohiuddine (2012) and many others. In recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces.…”

Generalized weighted random convergence in probability

“…al. [3]), (ii) k-statistical convergence of order a by Çolak and Bektas ß [5], (iii) weighted statistical convergence by Mursaleen et. al.…”

“…The main object of this paper is to modify the definition of weighted k-statistical convergence and to establish some important theorems related to the modes of convergences (a) to (d), which will effectively extend and improve all the existing results in this direction ( [1][2][3][4][5][6][8][9][10][11][12][13][14][15][16][17][22][23][24][25][26][27]). Moreover, intend to establish the relations among these four summability notions.…”