Lacunary statistical convergence. (English) Zbl 0794.60012

Fridy, J. A.; Orhan, C.

Pac. J. Math. 160, No. 1, 43-51 (1993).

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

Keywords:

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