Rough asymptotically deferred weighted statistical equivalence of order ? in probability and its applications to approximation theory

Abstract

The study of approximation theory and the asymptotic behavior of random variables are conventionally predicated on the assumption of classical convergence. Nevertheless, the attainment of classical convergence to a unique limit is frequently impeded in various physical and stochastic processes by measurement errors or inherent system roughness. To mitigate this issue, we introduce the concept of rough asymptotically deferred weighted statistical equivalence of order ? in probability. This novel structure generalizes classical asymptotic equivalence through the incorporation of a roughness degree r. We further define the notion of minimal roughness degree and scrutinize the algebraic properties of this new relation such as convexity. Moreover, we establish a rough Korovkin type approximation theorem for sequences of positive linear operators and provide an estimate regarding the rate of convergence. The manuscript concludes by presenting a numerical simulation to visualize our findings which serves to demonstrate strictly stronger generalizations of existing theories. © 2026 by the authors; licensee PSRP, Lahore, Pakistan.