On the Glivenko-Cantelli theorem for real-valued empirical functions of stationary $\alpha$-mixing and $\beta$-mixing sequences

TL;DR

This paper extends the classical Glivenko-Cantelli theorem to dependent sequences with $ ext{alpha}$- and $ ext{beta}$-mixing, providing conditions for empirical functions to converge uniformly under dependence.

Contribution

It introduces new sufficient conditions for the GC property in dependent sequences, relaxing independence assumptions and emphasizing the impact of mixing rates.

Findings

Established deviation bounds under mixing conditions.

Identified function classes satisfying uniform entropy conditions.

Extended GC theorem to $ ext{alpha}$- and $ ext{beta}$-mixing sequences.

Abstract

In this paper we extend the classical Glivenko-Cantelli theorem to real-valued empirical functions under dependence structures characterised 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 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.