Convergence of multivariate ECDF Gilvenko-Cantelli assures uniform a.s. convergence of univariate ECDF.  My questions are:


*

*Are there similar assurances for multivariate ECDF?  

*How is the rate of convergence dependent on the dimensions of the r.v.?


Thank you in advance.
 A: An answer to this question can be found here. In case you cannot get access to the book that is linked, I will repeat the result here:

Suppose $\mathcal{C}$ is a VC class of $\mathcal{X}$. Define,
$$ D_n(\mathcal{C})\equiv ||P_n - P||_\mathcal{C}$$
Then for all $\epsilon>0$,
$$\sup_P \Pr[\max_{m\geq n} D_m(\mathcal{C})\geq\epsilon]=\sup_P \Pr[\max_{m\geq n} ||P_m - P||\geq\epsilon] {\to} 0$$ as $n\to\infty$. That is, $$D_n(\mathcal{C})\overset{a.s.}{\to}0$$ uniformly over $P$.

As noted in the answer we can set $\mathcal{X}=\mathbb{R}^d$ to get at your question. The main motivating condition that we have is that $\mathcal{C}$ must be a VC class. Suppose we take $\mathcal{C}=\{(-\infty,x] : x \in \mathbb{R}^d\}$. Then we will find that the VC-index number of $\mathcal{C}$ is $v_\mathcal{C} = d+1$. For $\mathcal{C}$ to be a VC class we need that $v_\mathcal{C}<\infty$. This should give you some sense of how dimensionality is at work here. For more details, you should try to get a hold of the book of course.
