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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.

##### MSC:
 40A05 Convergence and divergence of series and sequences 40D05 General theorems on summability 40C05 Matrix methods for summability 11B05 Density, gaps, topology
##### Keywords:
lacunary sequence; statistical convergence; Cauchy sequence
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