Weighted Statistical Equivalence in Probability for Comparing Positive Linear Approximation Processes

Abstract

We introduce the notion of asymptotically weighted statistical equivalence of order alpha in probability for sequences of positive linear operators and study it within a Korovkin-type approximation framework. The emphasis is not only on the convergence of a single operator sequence to a target function, but on the asymptotic comparison of two approximation processes in a weighted probabilistic setting. After establishing the basic structural properties of the proposed equivalence relation and clarifying its connection with weighted statistical convergence in probability and with the classical statistical setting, we prove a Korovkin-type comparison theorem for normalized reference schemes. More precisely, we show that if a normalized sequence of positive linear operators has asymptotically small second moments and if two operator sequences are asymptotically weighted statistically equivalent in probability on the Test functions 1, x, and x2, then this equivalence extends to all functions in C[0, 1]. We also derive quantitative estimates for the deviation between the two approximation processes by means of the modulus of continuity and for functions in the Lipschitz class. The theory is illustrated by examples based on Bernstein operators, including a comparison with genuinely distinct positive linear schemes, showing that the present framework applies beyond the classical uniform approximation setting.