(Cross-listed from the math stackexchange)

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!

(Cross-listed from the math stackexchange)

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!