On statistical convergence and strong Cesàro convergence by moduli for double sequences

Abstract

A remarkable result on summability states that the statistical convergence and the strong Cesàro convergence are closely connected. Given a modulus function f, we will establish that a double sequence that is f-strong Cesàro convergent is always f-statistically convergent. The converse, in general, is false even for bounded sequences. However, we will characterize analytically the modulus functions f for which the converse of this result remains true. The results of this paper adapt to several variables the results obtained in (León-Saavedra et al. in J. Inequal. Appl. 12:298, 2019).