On statistical convergence. (English) Zbl 0588.40001

Fridy, J. A.

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)

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