Statistical properties of neutrosophic n-normed spaces in a non-Archimedean framework

Abstract

The notion of statistical convergence is a natural extension of ordinary convergence which has been studied by several authors in different spaces. Recently, several studies have shown a growing interest in convergence theory in neutrosophic spaces. In this research article, we investigate the statistical convergence and related notions of sequences in a non-Archimedean neutrosophic n-normed space. By combining natural density with the three neutrosophic components of truth-membership, indeterminacy-membership, and falsity-membership, we formulate statistical convergence with respect to the neutrosophic n-norm structure. Several equivalent characterizations of statistical convergence are established, including formulations in terms of exceptional index sets of natural density zero and componentwise statistical limits. We prove that the statistical limit, whenever it exists, is unique. Moreover, we show that ordinary convergence with respect to the neutrosophic n-norm implies statistical convergence, and we examine the corresponding behavior of algebraic operations, particularly the preservation of statistical convergence under addition. Statistical Cauchy sequences are also introduced and studied. In particular, every Cauchy sequence in the neutrosophic n-norm sense is shown to be statistically Cauchy. The notion of statistical completeness is then discussed through statistically Cauchy sequences and their statistical limits. Finally, illustrative examples in finite-dimensional real spaces are provided to clarify the abstract definitions and demonstrate sequences that are statistically convergent even when ordinary convergence fails. These results extend classical statistical convergence concepts to a neutrosophic non-Archimedean setting. The developed framework also provides a foundation for investigations of summability, approximation, and nonlinear analytical problems.