A. R. Freedman, J. J. Sember and M. Raphael [Proc. Lond. Math. Soc., III. Ser. 37, 508–520 (1978; Zbl 0424.40008)] presented a definition for lacunary refinement as follows: is called a lacunary refinement of the lacunary sequence if . They use this definition to present a one-side inclusion with respect to the refined and nonrefined sequence. 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 --convergence. For details, we refer the reader to the paper.