On Statistical Convergence

Abstract

The sequence X is statisticallv convergent to L if for eacli e > 0, lim" number o{ k < ti : | -L | > e} = 0; a: is a statisticallv Cauchv sequence if for each € > 0 there is a positive integer N = N{e) such thatlim" -{the number of k < n : | -| > t} = 0. n These concepts are shown to be equivaleot. Also, Statistical convergence is studied as a regulär summability method, and it is shown that it cannot be included by any matrix metbod. There are two Tauberian theorems proved: one uses the Tauberian condition Axi, = 0(1/Ar), which is shown to be "best possible," and the other is concerned with gap sequences.AMS 1980 Classification numbers: 40A05, 40C05, 40D25, 40EX)5 FridyRecall (see, e.g., [2, p. 290]) that the "natural density" of a set K of positive integers is defined bywhere | < n : keK} \ denotes the number of elements of K not exceeding n. We shall be particularly concerned with integer sets having natural density zero. To facilitate this we introduce the following notation: if x is a sequence such that Xj^ satisfies property P for all k except a set of natural density zero, then we say that a:^ satisfies P for "almost all k ", and we abbreviate this by "a.a. k."

I: Fréchet-Urysohn spaces

“…But in general, s-convergent sequences satisfy many of the properties of ordinary convergent sequences in metric spaces. It has been discussed and developed by many authors [3,5,6,9,10,11,21,22,25,26].…”

Lacunary generalized difference statistical convergence in random 2-normed spaces

“…A lot of developments have been made in this areas after the works ofSalát [26], Fridy [8] and Miller [21]. Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory and number theory.…”

Multiplier sequence spaces of fuzzy numbers dened by a Musielak-Orlicz function

“…The concept of statistical convergence was introduced by Fast [6] and also independently by Buck [3] and Schoenberg [18] for real and complex sequences. Further this concept was studied by Fridy [7,Connor [4]] and many others. Statistical convergence is closely related to the concept of convergence in Probability.…”