A measure theoretical subsequence characterization of statistical convergence

Abstract

Abstract.The concept of statistical convergence of a sequence was first introduced by H. Fast. Statistical convergence was generalized by R. C. Buck, and studied by other authors, using a regular nonnegative summability matrix A in place of C\ .The main result in this paper is a theorem that gives meaning to the statement: S= {sn} converges to L statistically (T) if and only if "most" of the subsequences of 5 converge, in the ordinary sense, to L . Here T is a regular, nonnegative and triangular matrix.Corresponding results for lacunary statistical convergence, recently defined and studied by J. A. Fridy and C. Orhan, are also presented.

Restricting statistical convergence

“…A sequence {x n } n∈N of real numbers is said to be statistically convergent to L if for arbitrary ε > 0, the set K(ε) = n ∈ N : |x n − L| ε has natural density zero. Statistical convergence turned out to be one of the most active areas of research in summability theory and some important developments on statistical convergence of single sequences can be seen from ( [4][5][6][7][8][9][10][11]). …”

Rates of Ideal Convergence for Approximation Operators

“…[6], [8], [10], [13]). Note that every convergent sequence is statistically convergent, but the converse is not always true.…”

“…A given sequence x = (x n ) is said to be A-statistically convergent to L if, for every ε > 0, δ A ({n : |x n − L| ≥ ε}) = 0, or equivalently, lim x n = L (see, e.g. [7], [10], [13]). Observe that if A = C 1 , then C 1 -statistical convergence coincides with statistical convergence.…”

Statistical convergence and measure convergence generated by a single statistical measure

“…In the recent years, various kinds of statistical convergence and their applications were extensively discussed in many pure and applied mathematical fields. As an example among a large amount of literature, we refer the reader to [3,15,16,22,23,25,26,30,36,37,42], and [49]. A sequence (x n ) in a topological space X is said to be statistically convergent to x ∈ X provided for any neighborhood U of x, we have χ A(ε) (j) = 0 ∞ (I) ⊂ ∞ and the quotient space ∞ / ∞ (I) for an ideal I.…”