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

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)?

• Would you mind clarifying "How does it impact improvements for these two types of convergence". For example, are you looking for a specific type of statement? Commented Apr 11, 2021 at 10:40
• As mentioned at en.wikipedia.org/wiki/…: "For every (fixed) $x$, $F_{n}(x)$ is a sequence of random variables which converge to $F(x)$ almost surely by the strong law of large numbers, that is, $F_{n}$ converges to $F$ pointwise. Glivenko and Cantelli strengthened this result by proving uniform convergence of $F_{n}$ to $F$." Commented Apr 11, 2021 at 10:43
• @SumanChakraborty I'm looking for any type of statement that improves the stochastic convergence (a.s. and/or in law) and is implied by the Glivenko-Cantelli theorem. Commented Apr 11, 2021 at 10:50
• Take a look at this paper (Theorem 1.3) dl.acm.org/doi/pdf/10.5555/3294771.3294929. This gives an error bound for Glivenko-Cantelli. If this is the kind of results that you are looking for, then I can point you to few more references. If you are looking for applications of GC then let me know as well, I can add some references. Best. Commented Apr 11, 2021 at 11:00
• @SumanChakraborty Thanks for the reference. Yes, some applications of the G-C would be helpful. Commented Apr 11, 2021 at 11:05

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.