Tauberian Theorems for the Statistical Convergence and the Statistical (C, 1, 1) Summability

Abstract

Every P-convergent double sequence is statistically convergent and every bounded statistically convergent sequence is statistical (C, 1, 1) summable.
The converse of these implications are not always true. Theorems on which conditioned converses are searched are known as Tauberian theorems.
Inspired by the convergence to zero of the difference sequence between a sequence and its arithmetic means in the single sequence case, we obtain Tauberian theorems for the statistical convergence and statistical (C, 1, 1) summability method by imposing some conditions on the difference sequence between a double sequence and its different arithmetic means.