Quantitative bounds for multivariate central limit theorem

Question

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,

Answer 1

There is a bunch of such statements which can be obtained by Stein's method.

You might be interested in the paper "On the Rate of Convergence in the Multivariate CLT" by Gotze, which is specifically devoted to Berry-Esseen theorems in the multidimensional setting. Have a look also at the very recent book Normal Approximation by Stein's Method by Chen, Goldstein and Shao.