On $A$-statistical convergence and $A$-statistical Cauchy via idea

Abstract

In [Analysis 1985, 5 (4), 301-313], J.A. Fridy proved an equivalence relation between statistical convergence and statistical Cauchy sequence. In this paper, we define $A^{I^{\ast }}$-statistical convergence and find under certain conditions, that it is equivalent to $A^{I}$-statistical convergence defined in [Appl. Math. Lett. 2012, 25 (4), 733-738]. Moreover, we define $A^{I}$-  and $A^{I^{\ast }}$-statistical Cauchy sequences and find some equivalent relation with $A^{I}$-  and $A^{I^{\ast }}$-statistical convergence.