Generalized almost statistical convergence

TL;DR

This paper introduces generalized almost statistical (GAS) convergence for bounded sequences, extending existing convergence notions, and explores its properties and the existence of related limit functionals.

Contribution

It defines GAS convergence, introduces Banach statistical limit functionals, and demonstrates sequences that are GAS convergent but not statistically or almost convergent.

Findings

Existence of Banach statistical limit functional.

Existence of GAS convergent sequences not convergent in other senses.

Topological properties of GAS convergent sequence space.

Abstract

The objective of this paper is to introduce the notion of generalized almost statistical (briefly, GAS) convergence of bounded real sequences, which generalizes the notion of almost convergence as well as statistical convergence of bounded real sequences. As a special kind of Banach limit functional, we also introduce the concept of Banach statistical limit functional and the notion of GAS convergence mainly depends on the existence of Banach statistical limit functional. We prove the existence of Banach statistical limit functional. Then we have shown the existence of a GAS convergent sequence, which is neither statistical convergent nor almost convergent. Also, some topological properties of the space of all GAS convergent sequences are investigated.