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ON THE GLIVENKO-CANTELLI THEOREM FORREAL-VALUED EMPIRICAL FUNCTIONS OF STATIONARY $\alpha$-MIXING AND $\beta$-MIXING SEQUENCES
Keywords:
$\alpha$-mixing, $\beta$-mixing, empirical process, stationarity, entropy number, law of large numbers, uniform convergence, Glivenko-Cantelli classDOI:
https://doi.org/10.17654/0972086325012Abstract
In this paper, we extend the classical Glivenko-Cantelli theorem to real-valued empirical functions under dependence structures characterized by α-mixing and β-mixing conditions. We investigate sufficient conditions ensuring that families of real-valued functions exhibit the Glivenko-Cantelli (GC) property in these dependence settings. Our analysis focuses on function classes satisfying uniform entropy conditions and establishes deviation bounds under mixing coefficients that decay at appropriate rates. Our results refine the existing literature by relaxing the independence assumptions and highlighting the role of dependence in empirical process convergence.
Received: May 15, 2025
Accepted: August 6, 2025
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