Distribution-free statistics on compact Lie groups

Question

(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!

Answer 1

From the perspective of statisticians, I will suggest two chapters in

Parthasarathy, Kalyanapuram Rangachari. Probability measures on metric spaces. Vol. 352. American Mathematical Soc., 1967.

Parthasarathy explained why it is natural to study probability measures on a Lie group. If you are looking forward to a more algebraic-flavored probability theory, this post might be of interest for you.

Your question is two-fold. One is to find a functional that measures the similarity between an empirical probability measure and another "True" measure; the other side is to discuss the asymptotic behavior of the functional w.r.t. empirical probability measures.

For the first problem, surprisingly there's relatively few functionals that measures the similarity between two probability measures on two (Banach) spaces. And most of such functionals are distances defined on $M(G)$, the space of probability measures defined on the group $G$ (and $G$ is taken as a manifold). One that I feel natural is the Wasserstein distance. The Kolmogorov-Smirnov distance is also one of them. If you are mainly interested in this fold of the problem, there is a recent reference

Chirikjian, Gregory S. Stochastic Models, Information Theory, and Lie Groups, Volume 2: Analytic Methods and Modern Applications. Vol. 2. Springer Science & Business Media, 2011.

For the second problem, since you are asking for the asymptotic behavior of such a empirical probability measure on a Lie group $G$, one approach is to find out an invariant measure that $F_n$ converges to, and therefore so does your statistical functional. For a general treatment of this approach, there is an influential book (especially chapter 2) that explained how you can apply Moore's result(Theorem 2.2.6) when you i.i.d. sample from $G_i$'s. I think there is a generalization to exchangeable case, but cannot remember for now. @Asaf pointed out that this theorem is for non-compact groups and hence might not be useful in your case, my ignorance.

Zimmer, Robert J. Ergodic theory and semisimple groups. Vol. 81. Springer Science & Business Media, 1984.

Another approach to study the asymptotic behavior of the empirical probability measures is the (notoriously hard, and forgive my stupidity) generic chaining technique. This approach, although does not claimed the asymptotic behavior of the empirical measures $F_n$, does use another measure (usually sub-Gaussian) to control the behavior of the sequence $F_n$.

Talagrand, Michel. "The generic chaining."Springer, 2005.

After clarifying two parts of your question, let me return to your question, the answer for Wasserstein distance mentioned above, is partially positive. However, I am doubtful that it can be extended to a very general compact Lie group without any clear product structure, even if semisimple. So if there is any reference, I would be interested to know.

del Barrio, Eustasio, Evarist Giné, and Carlos Matrán. "Central limit theorems for the Wasserstein distance between the empirical and the true distributions." Annals of Probability (1999): 1009-1071.