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