(Cross-listed from the [math stackexchange][1])

Let $(X_i)_{i=1}^n$ be iid random variables with joint cdf $F$. Recall that the *empirical distribution function* is:
$$
F_n(x) = \frac{1}{n} \sum_{i=1}^n \chi_{[-\infty,x]}(X_i) .
$$
Note that $F_n$ is a function-valued random variable. It is known that the Kolmogorov–Smirnov statistic:
$$
\mathrm{KS}_n = \|F-F_n\|_\infty ,
$$
converges almost surely to zero. Even better, if $F$ is continuous, the normalized statistic:
$$
K_n = \sqrt{n}\cdot \mathrm{KS}_n 
$$
converges to the Kolmogorov distribution (sup of the Brownian Bridge), which doesn't depend on the distribution of the $X_i$. 

> Let $G$ be a compact Lie group (semisimple, if that helps). Is there a similar statistic measuring how close a sequence in $G$ is to a given probability distribution, such that the limiting statistic is distribution-free?

Any ideas or pointers to a reference would be appreciated!


  [1]: http://math.stackexchange.com/questions/2015956/distribution-free-statistics-on-compact-lie-groups