Rough I -γτ-Statistical Convergence of Order α in Neutrosophic Normed Spaces

Abstract

In this paper, we investigate rough $I$ - $\gamma \tau$ -statistical convergence of order $\alpha$ in neutrosophic normed spaces and establish several fundamental structural properties of the associated limit sets. We first show that the family $\{L_j{\}}_{j\ge 0}$ of rough statistical limit sets is monotone with respect to the roughness parameter $j$ , and each $L_j$ is convex under mild monotonicity assumptions on the neutrosophic components. For $j=0$ , rough convergence reduces to the classical $I$ - $\gamma \tau$ -statistical convergence of order $\alpha$ , ensuring that the limit set is a singleton. We further demonstrate that the rough limit set is always neutrosophically closed and neutrosophically convex, highlighting its stability under both topological and geometric operations. A characterization of strong $I$ - $s{t}_{\gamma \tau }^{\alpha }$ -boundedness is obtained via the non-emptiness of the rough limit set. In addition, we introduce the notion of $I_j-s{t}_{\gamma \tau }^{\alpha }$ -cluster points and prove that every rough limit point is a cluster point, while the cluster set remains neutrosophically closed. Finally, we show that this convergence framework unifies several classical notions of statistical and ideal convergences as particular cases.