Statistical Convergence of Double Sequences on Probabilistic Normed Spaces

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 example 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.

On ideal convergence of double sequences in probabilistic normed spaces

“…Karakus and Demirci [20] defined the statistical analogues of convergence and Cauchy for double sequences on PN spaces with the help of double natural density given by Mursaleen and Ossama [5] as follows.…”

“…Recently Karakus et al [20] proved that in a PN space (X, ν, T ), a double sequence x = (x ij ) is statistically convergent to ξ if and only if there exists a subset K ⊂ N × N such that δ 2 (K) = 1 and ν-lim (i,j)∈K, i,j→∞ x ij = ξ. We use this well-known result of statistical convergence to introduce the notion of -convergence in PN spaces as follows.…”

“…For an extensive view on this subject, we refer [13][14][15][16][17][18][19]. Quite recently, Karakus et al [20] defined statistical analogues of convergence and Cauchy double sequences on PN spaces and gave a useful characterization. Subsequently, V. Kumar and K. Kumar [21] studied -Cauchy and -Cauchy sequences on the same spaces.…”

Lacunary generalized difference statistical convergence in random 2-normed spaces

“…Karakus [17] studied the concept of statistical convergence in probabilistic normed spaces. The theory of probabilistic normed(or metric) spaces was initiated and developed in [1], [28], [29], [30], [31] and further it was extended to random/probabilistic 2-normed spaces by Goleţ [13] using the concept of 2-norm which is defined by Gähler [11], and Gürdal and Pehlivan [15] studied statistical convergence in 2-Banach spaces.…”

On I-convergence in random 2-normed spaces

“…One of its generalizations is the ideal convergence or I-convergence which was introduced by Kostyrko et al [11] and studied by Balcerzak et al [2], Das et al [3], and Komisarski [12]. Recently, Karakus [10] studied the concept of statistical convergence in probabilistic normed spaces.…”