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

##### MSC:
 40A05 Convergence and divergence of series and sequences 40D05 General summability theorems 40C05 Matrix methods in summability 40E05 Tauberian theorems, general