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Lacunary statistical convergence. (English) Zbl 0794.60012
The sequence $$x$$ is statistically convergent to $$L$$ provided that for each $$\varepsilon>0$$, $\lim_ n {1 \over n} \{\text{the number of } k \leq n:| x_ k-L | \geq \varepsilon\}=0.$ A related concept is introduced by replacing the set $$\{k:k \leq n\}$$ with $$\{ k:k_{r-1}<k \leq k_ r\}$$, where $$\{k_ r\}$$ is a lacunary sequence, i.e., an increasing sequence of integers such that $$k_ 0=0$$ and $$\lim_ r(k_ r-k_{r-1})=\infty$$. The resulting summability method is compared to statistical convergence and to other summability methods, and questions of uniqueness of the limit value are considered.
Reviewer: J.A.Fridy

##### MSC:
 60F05 Central limit and other weak theorems 40G99 Special methods of summability
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