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Lacunary statistical convergence of double sequences. (English) Zbl 1106.40002
{\it A. R. Freedman}, {\it J. J. Sember} and {\it M. Raphael} [Proc. Lond. Math. Soc., III. Ser. 37, 508--520 (1978; Zbl 0424.40008)] presented a definition for lacunary refinement as follows: $\rho= \{\overline k_r\}$ is called a lacunary refinement of the lacunary sequence $\theta= \{k_r\}$ if $\{k_r\}\subseteq\{\overline k_r\}$. They use this definition to present a one-side inclusion with respect to the refined and nonrefined sequence. {\it J. Li} [Int. J. Math. Math. Sci. 23, 175--180 (2000; Zbl 0952.40001)] presented the other side of the inclusion. In this paper, the authors present a multidimensional analogue to the notion of refinement of lacunary sequences, and use this definition to present both sides of the above inclusion. In addition, the authors present a notion of double lacunary statistically Cauchy convergence and use this definition to establish that it is equivalent to the $S_{\theta r,s}$-$P$-convergence. For details, we refer the reader to the paper.

40A05Convergence and divergence of series and sequences
40G05Cesàro, Euler, Nörlund and Hausdorff methods
40C05Matrix methods in summability