Asymptotically deferred weighted statistical equivalence of order α in probability and its applications to approximation theory

Abstract

In this paper, we introduce a unified framework for studying the asymptotic behavior of random variables by combining deferred methods with weighted statistical convergence in probability. We define the notions of asymptotically deferred weighted statistical equivalence of order α in probability and asymptotically deferred weighted strong equivalence of order α in probability. We also establish the relationship between these two concepts and investigate the inclusion relations corresponding to different orders. In addition, we apply the proposed framework to approximation theory by establishing a Korovkin-type theorem for sequences of positive linear operators on C[0, 1]. Moreover, we present a Voronovskaya-type result describing the asymptotic behavior of the approximation error and derive an estimate for the rate of convergence by means of the modulus of continuity.