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Zbl 0794.60012
Fridy, J.A.; Orhan, C.
Lacunary statistical convergence. (English)
[J] Pac. J. Math. 160, No.1, 43-51 (1993). ISSN 0030-8730

The sequence $x$ is statistically convergent to $L$ provided that for each $\varepsilon>0$, $$\lim\sb n {1 \over n} \{\text {the number of } k \le n:\vert x\sb k-L \vert \ge \varepsilon\}=0.$$ A related concept is introduced by replacing the set $\{k:k \le n\}$ with $\{ k:k\sb{r-1}<k \le k\sb r\}$, where $\{k\sb r\}$ is a lacunary sequence, i.e., an increasing sequence of integers such that $k\sb 0=0$ and $\lim\sb r(k\sb r-k\sb{r-1})=\infty$. The resulting summability method is compared to statistical convergence and to other summability methods, and questions of uniqueness of the limit value are considered.
[J.A.Fridy]

MSC 2000:: *60F05 Weak limit theorems
40G99 Special methods of summability

Keywords: lacunary sequence; strongly almost convergent; statistical convergence; summability methods

Cited in: Zbl 1204.40003 Zbl 1183.46070 Zbl 1199.40025 Zbl 1176.46006 Zbl 1155.40301 Zbl 1109.40002 Zbl 1002.40001 Zbl 0952.40001 Zbl 0835.43006

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