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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)?

Disjoint Union
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  • 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? – Suman Chakraborty Apr 11 '21 at 10:40
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    As mentioned at https://en.wikipedia.org/wiki/Glivenko%E2%80%93Cantelli_theorem#Statement: "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 $F$." – Minus One-Twelfth Apr 11 '21 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. – Disjoint Union Apr 11 '21 at 10:50
  • Take a look at this paper (Theorem 1.3) https://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. – Suman Chakraborty Apr 11 '21 at 11:00
  • @SumanChakraborty Thanks for the reference. Yes, some applications of the G-C would be helpful. – Disjoint Union Apr 11 '21 at 11:05

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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).

Suman Chakraborty
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