Fridy, J. A. On statistical convergence. (English) Zbl 0588.40001 Analysis 5, 301-313 (1985). A sequence \(\{x_ k\}^{\infty}_{k=1}\) is said to be statistically convergent to \(L\) provided that the density of the set \(\{k\in\mathbb N: | x_ K-L| \geq \varepsilon \}\) is 0 for each \(\varepsilon >0\) (the density of the set \(M\subset N\) is the number \(\lim_{n\to \infty}M(n)/n\), where \(M(n)\) denotes the number of elements of \(M\) not exceeding \(n\)). The author gives an equivalent condition of Cauchy type for the statistical convergence. This convergence can be regarded as a regular summability method. This method cannot be included by any matrix summability method. Two Tauberian conditions are given for the statistical convergence. One of them is the following: \(\Delta x_ k=O(1/k)\). Reviewer: Tibor Šalát (Bratislava) Cited in 11 ReviewsCited in 595 Documents MSC: 40A05 Convergence and divergence of series and sequences 40D05 General theorems on summability 40C05 Matrix methods for summability 40E05 Tauberian theorems Keywords:sequence; statistical convergence; regular summability method PDF BibTeX XML Cite \textit{J. A. Fridy}, Analysis 5, 301--313 (1985; Zbl 0588.40001) Full Text: DOI