Let $\mu_n$ be an empirical distribution of $n$-iid points from the underlying distribution $\mu$.
In 1D, it is well-known by Kolmogorov's theorem, Glivenko–Cantelli theorem that for any $x$, let $E_x = (-\infty, x]$. Then $$ \lim_{n\to \infty} \sup_{x \in \mathbb{R}}|\mu_n(E_x)-\mu(E_x)| = 0 \quad (a.s.) $$ and $$ \lim_{n\to \infty} \sqrt{n}\sup_{x\in \mathbb{R}}|\mu_n(E_x) - \mu(E_x)| \overset{d}{=}\sup_{t \in [0,1]} |B(t)| $$ where $B(t)$ is the Brownian bridge. In $d$-dimension, are there analogous results for this? If yes, I am looking for a reference of this result.
Any comments/answers/suggestions will be very appreciated.
The concept of VC-Dimension precisely generalizes this phenomenon. If a set of functions has finite VC Dimension, such statments would hold. In higher dimensions you can look at the set of indicator functions of cuboids $ \mathcal{F} = \{ E_{x_1,\dots,x_d} : x_1, \dots, x_d \in \mathbb{R} \} $ where $E_{x_1,\dots,x_d}(y_1,\dots,y_d) = \prod_{i=1}^d E_{x_i}(y_i)$. It can be shown $\mathcal{F}$ has finite VC-Dimension. (I don't remember exactly but somewhere between $d$ and $2d$).
You can consult Chapter 8 of HDP[1] for the first kind of convergence results. For a Donsker Theorem, you can consult Shorack and Wellner[2].
[1]: High-Dimensional Probability, Roman Vershynin. https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-book.html
[2]: Empirical Processes with Applications to Statistics, Galen R. Shorack and Jon August Wellner.