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Lacunary statistical convergence of multiple sequences. (English) Zbl 1132.40312
Summary: Quite recently, Mursaleen and O. H. H. Edely [J. Math. Anal. Appl. 288, No. 1, 223–231 (2003; Zbl 1032.40001)], defined the statistical analogue for double sequences $$x=\{x_{k,l}\}$$ as follows: A real double sequence $$x=\{x_{k,l}\}$$ is said to be P-statistically convergent to $$L$$ provided that for each $$\epsilon >0$$
$P-\lim_{m,n} \frac{1}{mn}\{\text{ numbers of } (j,k):j<m \text{ and } k<n, |x_{j,k}-L|\geq\epsilon\}.$
In this paper we introduce and study lacunary statistical convergence for double sequences and we shall also present some inclusion theorems.

##### MSC:
 40G99 Special methods of summability 42B15 Multipliers for harmonic analysis in several variables 40C05 Matrix methods for summability
##### Keywords:
double lacunary sequences; $$p$$-convergent
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##### References:
 [1] Fast, H., Sur la convergence statistique, Colloq. math., 2, 241-244, (1951) · Zbl 0044.33605 [2] Freedman, A.R.; Sember, J.J.; Raphael, M., Some Cesàro type summability spaces, Proc. London math. soc., 37, 508-520, (1978) · Zbl 0424.40008 [3] Fridy, J.A.; Orhan, C., Lacunary statistical convergent, Pacific J. math., 160, 1, 43-51, (1993) · Zbl 0794.60012 [4] Hamilton, H.J., Transformations of multiple sequences, Duke math. J., 2, 29-60, (1936) · Zbl 0013.30301 [5] M. Mursaleen, O.H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. (in press) · Zbl 1032.40001
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