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

### MSC:

 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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