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](https://en.wikipedia.org/w/index.php?title=Central_limit_theorem&oldid=397609863#Multidimensional_central_limit_theorem))


This question is a re-post from


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


Thanks,



Quantitative bounds for multivariate central limit theorem
There is a bunch of such statements which can be obtained by [Stein's method](https://en.wikipedia.org/wiki/Stein%27s_method).


You might be interested in the paper ["On the Rate of Convergence in the Multivariate CLT"](https://projecteuclid.org/euclid.aop/1176990448) 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*](https://books.google.co.uk/books?id=5jMVpAbs9UkC&printsec=frontcover&dq=Normal+Approximation+by+Stein%2527s+Method&hl=en&ei=UFbsTPL-FcShOqrIzJEB&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCkQ6AEwAA#v=onepage&q&f=false) by Chen, Goldstein and Shao.


