The celebrated BerryEsseen 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 2dimensional 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?
1 Answer
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, 170203, 304, Trudy Inst. Mat., 20, Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 1993
that you have the Edgeworth expansion of the form $$P(\V_n\^2\le r)=P(\G\^2\le r)+\sum_{1\le m\le(k2)/2}n^{m} Q_{2m}(r)+\Delta_{n,k}(r),$$ where $G$ is a meanzero 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 $1/n$.

$\begingroup$ Thank you so much! Exactly the thing I was looking for! $\endgroup$– TOMMar 8, 2019 at 13:34