Glivenko-Cantelli classes for real-valued empirical functions of stationary $α$-mixing and $β$-mixing sequences

The Glivenko-Cantelli theorem establishes uniform convergence of empirical distribution functions to their theoretical counterpart. This paper extends the classical result to real-valued empirical functions under dependence structures characterized by $\alpha$-mixing and $\beta$-mixing conditions. We investigate sufficient conditions ensuring that families of real-valued functions exhibit the Glivenko-Cantelli (GC) property in these dependent settings. Our analysis focuses on function classes satisfying uniform entropy conditions and establishes deviation bounds under mixing coefficients that decay at appropriate rates. Our findings refine existing literature by relaxing independence assumptions and highlighting the role of dependence in empirical process convergence.