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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.

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