Does a DKW-type inequality hold for the empirical CDF of a random vector on the sphere?

Question

I've begun to study concentration of measure because of its relevance to statistical mechanics. In recent decades concentration inequalities have played a role in elucidating foundational conceptual issues such as the degree to which equilibrium states are typical in the state space of physical systems (see e.g. this paper).

As far as I can tell as someone only newly introduced to concentration of measure, a lot is known about concentration for certain classes of sufficiently nice functions $f:S^{N-1}\to\mathbb R$ (Levy's Lemma for example). A lot also seems to be known about sequences $X_1, \dots, X_N$ of i.i.d. random variables (the DKW inequality for example). But I'm interested in results that don't seem to follow, at least in a way that's sufficiently straightforward to me, from standard concentration inequalities, and I'm having trouble finding what's known about such statements. Here's an example that I had originally thought would be simple to find in the literature:

Let $X = (X_1, \dots, X_N)$ be a uniform random vector on $S^{N-1}$. The $X_i$ are not independent, but I have a sense that they should be approximately independent for large $N$, so I suspect that something like the DKW inequality should hold in this case. Perhaps something like this: Let $F_N$ be the corresponding empirical CDF; \begin{align} F_N(x) = \frac{1}{N} \sum_{i=1}^N \mathbf 1_{\{X_i \leq x\}} \end{align} and let $F$ be the marginal CDF of each $X_i$ (which I think is asymptotically gaussian), then for ever $\epsilon > 0$, \begin{align} \mathbf P\left(\sup_{x\in \mathbf R}\left|F_N(x) - F(x)\right| > \epsilon\right) \leq \text{something that's small for large $N$} \end{align} Is there such a result? If so, does it follow from well-known concentration theory?

Answer 1

[Too long for a comment, yet not a full answer]

The trick in this particular example should be to use the fact that one can construct $X$ as $Z/\|Z\|$, where $Z$ is a Gaussian random vector with identity covariance matrix. One thus has $$ \frac1N \sum_{i=1}^N {\bf{1}}\{X_i \le x \} = \frac1N \sum_{i=1}^N {\bf{1}}\{Z_i \le x\|Z\| \} . $$ Furthermore, $\|Z\|$ will concentrate around $\sqrt{N}$. I guess that here, the correct object to look at is thus $F_N(x/\sqrt{N})$. The remaining issue is to control the difference $$ \frac1N \sum_{i=1}^N {\bf{1}}\{Z_i \le x \} - \frac1N \sum_{i=1}^N {\bf{1}}\left\{Z_i \le x\ \sqrt{\frac{\|Z\|^2}{N}} \right\}. $$ As, by the properties of chi-square and the delta method, $$ \sqrt{\frac{\|Z\|^2}{N}} = 1 + O_p(N^{-1/2}), $$ the difference in the display above won't play any role for the result you mentioned above. If you look for precise concentration, it might well be that you get additional fluctuations.