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

TL;DR

This work extends the classical Glivenko-Cantelli theorem to real-valued empirical function classes under dependent data modeled by $α$-mixing and $β$-mixing. Building on the Sangaré–Lo functional GC framework, it derives GC results for stationary sequences by combining entropy (bracketing) and VC-type conditions with explicit decay rates on mixing coefficients. The main contributions include a general GC theorem for arbitrary stationary sequences, GC results for real-valued empirical functions under $α$-mixing and $β$-mixing with decay rates $α(n)=O(n^{-(1+δ)/(1-δ)})$ and $β(n)=O(n^{-(1+δ)/(1-δ)})$, and the extension to function classes beyond indicator-based empirical distributions. These results provide theoretical guarantees for uniform convergence in dependent data settings, with implications for time series analysis and statistical learning under dependence.

Abstract

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

Theorems & Definitions (21)

• Theorem 1
• Remark 1
• Theorem 2
• Remark 2
• Theorem 3
• Theorem 4
• Lemma 1
• proof
• proof
• Theorem 5