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$$\bar {\lambda }$$-statistically convergent double sequences in probabilistic normed spaces. (English) Zbl 1274.40016
The notion of $$\lambda$$-statistical convergence was introduced by M. Mursaleen [Math. Slovaca 50, 111–115 (2000; Zbl 0953.40002)] and was further studied by A. Alotaibi in probabilistic normed space [Open Math. J. 1, 82–88, (2008; Zbl 1177.46056)]. In this paper, the authors extend this concept for double sequences in probabilistic normed spaces and call it $$S_{\bar {\lambda }}^{(PN)}$$-convergence. They prove that convergence in probabilistic norm implies the $$S_{\bar {\lambda }}^{(PN)}$$-convergence, obtain the condition for implications of $$S_{\bar {\lambda }}^{(PN)}$$-convergence from statistical convergence of double sequences, and also establish the subsequence characterization for $$S_{\bar {\lambda }}^{(PN)}$$-convergence. Furthermore, they introduce the idea of $$\bar {\lambda }$$-statistical Cauchy double sequences in PN-spaces, naming it $$S_{\bar {\lambda }}^{(PN)}$$-Cauchy, and obtain its relation with $$S_{\bar {\lambda }}^{(PN)}$$-convergence.

##### MSC:
 40A35 Ideal and statistical convergence 40J05 Summability in abstract structures 46S50 Functional analysis in probabilistic metric linear spaces
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