How does the Glivenko-Cantelli theorem improve the stochastic convergence of the empirical distribution $F_n(x)$?

Question

Let $X_i$ be iid random variables with empirical cumulative distribution function $F_n(x)$ and CDF $F(x)$. From the central limit theorem and the strong law of large numbers, we know that $F_n\stackrel{d/a.s.}{\to}F$. The Glivenko-Cantelli theorem states that $\sup\limits_{x\in\mathbb R}|F_n(x)-F(x)|\to 0$ almost surely. How does it impact improvements for these two types of convergence (by itself or maybe by other theorems that are implied)?

Answer 1

Let me refer you to two applications:

1. Statistics: theory of empirical process has been widely applied in statistics (especially non-parametric). I have just found these notes on the internet. Moreover, there are at least two textbooks on this topic. Introduction to Empirical Processes and Semiparametric Inference by Kosorok, and Weak Convergence and Empirical Processes by van der Vaart and Wellner.

2. Probability and Combinatorics: there are many applications but I have found the application to the $K$-core problem to be cute. I strongly recommend this paper: A simple solution to the k-core problem by Janson and Lucjak.

I have used it to study bootstrap percolation here (section 3.2).