Abstract
The concept of statistical convergence was presented by Steinhaus in 1951. This concept was extended to the double sequences by Mursaleen and Edely in 2003. Karakus has recently introduced the concept of statistical convergence of ordinary (single) sequence on probabilistic normed spaces. In this paper, we define statistical analogues of convergence and Cauchy for double sequences on probabilistic normed spaces. Then we display an exampl e such that our method of convergence is stronger than usual convergence on probabilistic normed spaces. Also we give a useful characterization for statistically convergent double sequences.
Highlights
An interesting and important generalization of the notion of metric space was introduced by Menger [1] under the name of statistical metric, which is called probabilistic metric space
The theory of probabilistic metric space was developed by numerous authors, as it can be realized upon consulting the list of references in [2], as well as those in [3, 4]
We extended in [5] the concept of statistical convergence from single to multiple sequences and proved some basic results
Summary
An interesting and important generalization of the notion of metric space was introduced by Menger [1] under the name of statistical metric, which is called probabilistic metric space. The theory of probabilistic metric space was developed by numerous authors, as it can be realized upon consulting the list of references in [2], as well as those in [3, 4]. An important family of probabilistic metric spaces are probabilistic normed spaces. The theory of probabilistic normed spaces is important as a generalization of deterministic results of linear normed spaces. The concept of statistical convergence of ordinary (single) sequence on probabilistic normed spaces was introduced by Karakus in [5]. We extended in [5] the concept of statistical convergence from single to multiple sequences and proved some basic results. We recall some notations and definitions which we use in the paper
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