Rate of decay in the multivariate Central Limit Theorem

Question

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?

Answer 1

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 $1/n$.