What is the order of the constant $K$ in the multidimensional Dvoretzky-Kiefer-Wolfowitz inequality($Ke^{-c z}$)?

Question

Let $F_n$ be the empirical distribution obtained from an i.i.d. sample of the distribution $F:R ^d \to [0, 1]$. Kiefer (1961) shows that the convergence of the empirical distribution is like $$ P\left( \left\lVert F_n - F\right\rVert_\infty > z \right) \le K \exp \left( - \left(2 - \epsilon\right) n z^2 \right) $$ where the constant $K$ depends on the dimension $d$ and the threshold $\epsilon >0$.

What is the order of the constant $K$?
Is it like $2^d$ or is it much smaller?

In the prood for the case $d=1$, I see that $K$ is approximatively the tail of some distribution, so it is small. For higher dimensions, the proof proceed by induction and I don't see how the value of $K$ evolves.

Definition of $F_n$:
For a vector $x$ in $R^d$, DKW defines the empirical distribution as the number of sample point $X_i$ satisfying $X_i < x$, coordinate-wise; divided by the number of sampled points $n$.

Reference:
DKW wiki
DKW d=1: Dvoretzky, Kiefer, Wolfowitz 1958
DKW d>1: Kiefer, Wolfowitz, 1958
DKW d>1, sharp exponent: Kiefer, 1961

Answer 1

For any distribution $F$ over the naturals $\mathbb{N}$, one can show the following DKW-type inequality: $$ \mathbb{P}(||F-F_n||_\infty > 1/\sqrt{n}+\epsilon) \le \exp(-2n\epsilon^2) $$ (in fact, it's known for the more general Markov case, see http://projecteuclid.org/euclid.jap/1421763330 )

I started to write something about $F$ being well-approximated by discrete distributions, but now I think the question may be ill-posed. How do you define the empirical $F_n$ in dimension $d>1$? In $R^1$ you can count how many sample points appeared to the left of $x$; what's the higher-dimensional analogue? For that matter, what's the analogue of a CDF in higher dimensions?

I'm leaving my answer because I think it's relevant for some reasonable reformulation of the question. The bottom line is that you should get dimension-free constants.

Answer 2

I recently found the constant is 2d asymptotically,

https://www.sciencedirect.com/science/article/pii/S016771522100050X

https://en.wikipedia.org/wiki/Dvoretzky%E2%80%93Kiefer%E2%80%93Wolfowitz_inequality

Answer 3

I wrote the paper and the answer is 2d, at least asymptotically. In my paper, I actually prove that Kiefer’s lower bound argument is wrong. There are a handful of papers that also reach the incorrect conclusion. Largely based on Kiefer’s faulty 1958 counter example.