The celebrated Berry-Esseen inequality tells us that the rate of convergence in the univariate CLT is of magnitude $\frac{1}{\sqrt{n}}$ for sums $S\_n=X\_1+\cdots+X\_n$ of independent random variables $X\_i$ with bounded third moments. Consider the following example: let $\varepsilon\_{i}$ stand for independent Rademacher random variables, that is, we have $\mathbb{P}(\varepsilon\_{i}=\pm 1)=1/2$. I am interested in the random vector $V\_n=(R\_n,R'\_n)$, where $R\_n=\varepsilon\_{1}+\cdots+\varepsilon\_{n}$ and $R'\_{n}$ is an independent copy of $R\_n$. It is easy to verify that for fixed $r\geq \sqrt{2}$ we have $\mathbb{P}(||V\_n||\leq r)\approx \frac{c\_{r}}{n}$ for some $c\_{r}>0$. I would like to estimate the proximity between the law of $V\_n$ and a correspoding 2-dimensional Gaussian distribution. For an estimate of the proximity that would make sense one would hope for a bound of magnitude at most $\frac{c}{n}$. Are such results known?



Rate of decay in the multivariate Central Limit Theorem
It follows from the main result in the paper 



> 
> MR1309710 (95k:60015) Nagaev, S. V.; Chebotarëv, V. I. On the
>  Edgeworth expansion in a Hilbert space. (Russian) Limit theorems for
>  random processes and their applications, 170--203, 304, Trudy Inst.
>  Mat., 20, Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk,
>  1993
> 
> 
> 


that you have the Edgeworth expansion of the form
$$P(\|V\_n\|^2\le r)=P(\|G\|^2\le r)+\sum\_{1\le m\le(k-2)/2}n^{-m} Q\_{2m}(r)+\Delta\_{n,k}(r),$$
where $G$ is a mean-zero Gaussian vector with the same covariance matrix as that of $V\_n$, $k\ge4$, $Q\_{2m}$ are certain functions, and $\Delta\_{n,k}(r)$ is a remainder. So, the main asymptotic term in the "error" $P(\|V\_n\|^2\le r)-P(\|G\|^2\le r)$ of the normal approximation is $Q\_2(r)/n$, indeed on the order of $1/n$. 


