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Lacunary statistical convergence and inclusion properties between lacunary methods. (English) Zbl 0952.40001
The integer sequence \(\theta=\{k_r\}\) is called a lacunary sequence if it is increasing and \(\lim_{r\to \infty} (k_r-k_{r-1})= \infty\). A complex number sequence \(x=\{x_k\}\) is said to be \(s_{\theta}\)-convergent to \(L\) if for each \(\varepsilon >0\) one has \[ \lim_{r\to \infty} \frac 1{k_r-k_{r-1}} \# \{k: k_{r-1}< k \leq k_r\text{ and } |x_k-L|\geq \varepsilon \} = 0. \] Let \(S_{\theta}\) be the family of all sequences \(x\) which are \(s_{\theta}\)-convergent to some \(L\). In this paper, which continues the work of J. A. Fridy and C. Orhan [Pac. J. Math. 160, No. 1, 43-51 (1993; Zbl 0794.60012)], the author studies inclusion properties between \(S_{\theta}\) and \(S_{\beta}\), where \(\theta\) and \(\beta\) are two arbitrary lacunary sequences.

40A05 Convergence and divergence of series and sequences
40D05 General theorems on summability
40C05 Matrix methods for summability
11B05 Density, gaps, topology
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