On statistical convergence in fractal analysis

Abstract

This study investigated the statistical convergence of fractal-generating set sequences, motivated by the observation that natural fractals, influenced by external biological, chemical, or physical factors, rarely exhibit strict classical convergence. Instead, their limiting behavior often aligns with statistical patterns. We formalized the concept of statistical convergence for compact subsets of Rn, introduced the notion of statistical Cauchy sequences, and established their sufficiency for statistical convergence-mirroring the classical relationship. Several illustrative examples and graphical simulations, including variants of the Sierpinski triangle and Koch snowflake, highlight the distinction between classical and statistical convergence. The proposed framework provides a more realistic and robust approach to understanding fractal structures in both theoretical and applied contexts.