Kolmogorov distance

Question

Can someone provide an example of direct me to an example of a problem where Kolmogorov distance is used. I think I understand the definition but would like to see an illustration to solidify my understanding and can't find anything. I am currently studying it as a way to measure the distance between a given distribution and the normal distribution.

Answer 1

In the classical Glivenko-Cantelli Theorem. It states the following: if $(X_n)_{n\geq 0}$ are iid random variables with common cdf $F(x)$ and $F_n(x)$ is the empirical cdf defined as $$ F_n(x)=\dfrac{1}{n}\sum_{j=1}^nI(X_i\leq x), $$ then, $$ \sup_{x\in\mathbb{R}}|F_n(x)-F(x)|\overset{a.s.}{\longrightarrow}0,\quad n\to\infty. $$ This states that $F_n(x)$ approaches $F(x)$ uniformly with probability one whenever the sample size tends to infinity, i.e., the Kolmogorov distance between $F_n(x)$ and $F(x)$ converges to zero with probability one.