Further results on I−deferred statistical convergence

Abstract

For a non-empty set X, an ideal I represents a family of subsets of X that is closed under taking finite unions and subsets of its elements. Considering X = N, in the present study, we set forth with the new notion of I−deferred statistical limit point, I−deferred statistical cluster point and study various properties of the newly introduced notion. For a real valued sequence x = (xn), we prove that every I−deferred statistical limit point is an I−deferred statistical cluster point. Moreover, the collection of all I−deferred statistical cluster points of x is a closed subset of R. We also introduce the notion of I−deferred statistical limit superior and inferior for real valued sequences and prove several interesting properties. In the end, we establish a necessary and sufficient condition under which a I−deferred statistically bounded real valued sequence is I−deferred statistically convergent.