On strong $\mathcal{A}^{\mathcal{I}}$-statistical convergence of sequences in probabilistic metric spaces

TL;DR

This paper explores advanced concepts of strong $ ext{A}^{ ext{I}}$-statistical convergence and Cauchyness in probabilistic metric spaces, introducing new types and analyzing their properties and relationships.

Contribution

It introduces and studies strong $ ext{A}^{ ext{I}}$-statistical convergence, Cauchyness, and limit points in probabilistic metric spaces, extending prior notions in the field.

Findings

Established properties of strong $ ext{A}^{ ext{I}}$-statistical convergence.

Defined and analyzed strong $ ext{A}^{ ext{I}^*}$-statistical Cauchyness.

Explored relationships between different statistical Cauchyness concepts.

Abstract

In this paper using a non-negative regular summability matrix $\mathcal{A}$ and a non-trivial admissible ideal $\mathcal{I}$ in $\mathbb{N}$ we study some basic properties of strong $\mathcal{A}^{\mathcal{I}}$-statistical convergence and strong $\mathcal{A}^{\mathcal{I}}$-statistical Cauchyness of sequences in probabilistic metric spaces not done earlier. We also introduce strong $\mathcal{A}^{\mathcal{I^*}}$-statistical Cauchyness in probabilistic metric space and study its relationship with strong A$\mathcal{A}^{\mathcal{I}}$-statistical Cauchyness there. Further, we study some basic properties of strong $\mathcal{A}^{\mathcal{I}}$-statistical limit points and strong $\mathcal{A}^{\mathcal{I}}$-statistical cluster points of a sequence in probabilistic metric spaces.