Lacunary statistical convergence and inclusion properties between lacunary methods.

*(English)*Zbl 0952.40001The integer sequence $\theta =\left\{{k}_{r}\right\}$ is called a lacunary sequence if it is increasing and ${lim}_{r\to \infty}({k}_{r}-{k}_{r-1})=\infty $. A complex number sequence $x=\left\{{x}_{k}\right\}$ is said to be ${s}_{\theta}$-convergent to $L$ if for each $\epsilon >0$ one has

$$\underset{r\to \infty}{lim}\frac{1}{{k}_{r}-{k}_{r-1}}\#\{k:{k}_{r-1}<k\le {k}_{r}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}|{x}_{k}-L|\ge \epsilon \}=0\xb7$$

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.

Reviewer: Laśzló Tóth (Pécs)