### 2005-15

#### Non-classical Bounds in the Central Limit Theorem in R^d

Series: 2005-15, Preprints

MSC:
60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks

Abstract:
Let $X_1, ..., X_n$ be independent
random vectors taking values in $R^d$ such that $\mathbf{E}X_k=0$,
for all $k$. Write $S=X_1+...+X_n$. Assume that the covariance
operator, say $C^2$, of $S$ is invertible. Let $Z$ be a centered
Gaussian random vector such that covariances of $S$ and $Z$ are
equal. Let $\sigma_j^2=\textrm{tr cov}(C^{-1}X_j), B^2_{1n}=\sigma_1^2+...+\sigma_n^2, L_{1n}=(\sigma_1^3+...+\sigma_n^3)\backslash B^3_{1n}$. Let
$\mathcal C$ stand for the class of all convex Borel subsets of
$R^d$. We get a bound for $\Delta=\sup_{A\in\mathcal C}|P\{S\in A\}-P\{Z\in A\}|$. Namely, $\Delta\leq Md^3(\nu_3+(\nu_3)^{\frac{1}{4}}(L_{1n})^{\frac{3}{4}}))$ with
$\nu_3=\nu_{31}+...+\nu_{3n}, \nu_{3k}=\int\limits_{R^d}|C^{-1}z|^3|d(F_k-\Phi_k(z)|$, where $M$ is
absolute constant. In the case of i.i.d. random vectors $X_1, ..., X_n$ we have $\Delta=O(1/\sqrt{n})$.}

Keywords:
rate of convergence, pseudomoments