Generalized statistical convergence and statistical core of double sequences

Some remarks on statistical summability of order $\tilde{\alpha}$ defined by generalized De la Vall\'{e}e-Pousin Mean

“…have shown their interest to study double sequences and related convergence problems. Mursaleen and Edely [15] and Mursaleen et al [14] respectively extended Definition 1.1 and Definition 1.2 on double sequences and obtained some analogous results. However, Çolak and Altin [4] introduced statistical convergence of orderα for these kind of sequences.…”

“…We begin with the following definition: Let Sα (λ,µ) (x) denotes the set of all (λ, µ)−statistically convergent double sequences of orderα. Forα = (a, b) = (1, 1), Definition 2.1 coincides with (λ, µ)−statistical convergence of double sequences of [14]. For the choice λ = (n) and µ = (m), Definition 2.1 coincides with statistical convergence of double sequences of orderα of [3].…”

“…Let x = (x ij ) and y = (y ij ) be two double sequences of complex numbers andα ∈ (0, 1]. [14]. For λ = (n) and µ = (m), Definition 2.5 coincides with strong p-Cesàro summability of double sequences of orderα of [3].…”

On $\Lambda$-statistical convergence of order $\alpha$ in random 2-normed space

“…The concept of statistical convergence was introduced by Fast [5] and Steinhaus [23] and then reintroduced by Schoenberg [20] independently. Over the years, statistical convergence has been developed in [3,6,7,12,16,17,24] and turned out very useful to resolve many convergence problems arising in Analysis.…”

On ideal convergence in probabilistic normed spaces

“…The concept of statistical convergence for sequences of real numbers was introduced by Fast [4] and Steinhaus [22] independently in the same year 1951 and since then several generalizations and applications of this notion have been investigated by various authors, namely [3], [7], [15], [16], [17], [18], [19]. One of its interesting generalization is I-convergence which was given by Kostyrko et al [12].…”