# Rate of decay in the multivariate Central Limit Theorem

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?

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 $$1/n$$.