Multivariate extension of the Glivenko-Cantelli Theorem Is there a multivariate extension of the Glivenko-Cantelli theorem for empiric process. If not, are there some weaker statements or other results about the probability limits of empirical distributions of vector of real (or integer) random variables?

There is a far reaching generalization by Vapnik and Chervonenkis. See theorem 1 on page 828 (chapter 26) in Shorack and Wellner's 1986 book on empirical processes. For the multivariate case dealing driectly with Euclidean spaces $\mathbb{R}^d$, you would be particularly interested to check examples 1-5 on pages 833-834 of the same book.