A Constructive Proof of the Glivenko-Cantelli Theorem

TL;DR

This paper provides a constructive proof of the Glivenko-Cantelli theorem, demonstrating how empirical distribution functions converge uniformly to the true distribution, reinforcing the validity of sampling for distribution inference.

Contribution

It offers a new constructive proof of the Glivenko-Cantelli theorem, directly deriving convergence from empirical distribution properties and enhancing understanding of sampling-based distribution analysis.

Findings

Empirical distribution functions converge uniformly almost surely to the true distribution.

The proof is constructive, providing explicit derivation of convergence.

Sampling techniques effectively capture distribution behavior as sample size increases.

Abstract

The Glivenko-Cantelli theorem states that the empirical distribution function converges uniformly almost surely to the theoretical distribution for a random variable $X \in \mathbb{R}$. This is an important result because it establishes the fact that sampling does capture the dispersion measure the distribution function $F$ imposes. In essence, sampling permits one to learn and infer the behavior of $F$ by only looking at observations from $X$. The probabilities that are inferred from samples $\mathbf{X}$ will become more precise as the sample size increases and more data becomes available. Therefore, it is valid to study distributions via samples. The proof present here is constructive, meaning that the result is derived directly from the fact that the empirical distribution function converges pointwise almost surely to the theoretical distribution. The work includes a proof of this preliminary statement and attempts to motivate the intuition one gets from sampling techniques when studying the regions in which a model concentrates probability. The sets where dispersion is described with precision by the empirical distribution function will eventually cover the entire sample space.