For the univariate central limit theorem, the Berry-Esseen theorem gives a quantitative bound on the rate of convergence of distributions to the Normal distribution under Kolmogorov distance:

https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem

Are similar statements known for the multivariate version of the central limit theorem, that use some standard distance measure?

https://en.wikipedia.org/wiki/Central_limit_theorem#Multidimensional_central_limit_theorem (current revisions)

This question is a re-post from

https://math.stackexchange.com/questions/11596/quantitative-bounds-for-multivariate-central-limit-theorem

Thanks,