On statistical convergence. (English) Zbl 0588.40001

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: \(\Delta x_ k=O(1/k)\).


40A05 Convergence and divergence of series and sequences
40D05 General theorems on summability
40C05 Matrix methods for summability
40E05 Tauberian theorems
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