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On statistical convergence. (English) Zbl 0588.40001
A sequence $\{x\sb k\}\sp{\infty}\sb{k=1}$ is said to be statistically convergent to L provided that the density of the set $\{$ $k\in N:$ $\vert x\sb K-L\vert \ge \epsilon \}$ is 0 for each $\epsilon >0$ (the density of the set $M\subset N$ is the number $\lim\sb{n\to \infty}M(n)/n$, 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: $\Delta x\sb k=O(1/k)$.
Reviewer: T.Šalát

40A05Convergence and divergence of series and sequences
40D05General summability theorems
40C05Matrix methods in summability
40E05Tauberian theorems, general