On approximation of convolutions by normal laws

Let $F_n$ be the distribution of $\xi_1+\dots+\xi_n$, where $\xi_i$ are independent random vectors with values in $R^k$; $G_n$ is the Gaussian distribution in $R^k$ with mean and covariances equal to those of $F_n$. Let $\mathfrak L_{\Pi}(F,G)$ be the Lévy–Prokhorov distance between $k$-dimensional distributions defined according to the norm $|\cdot|$ in $R^k$.
The main result of the paper is the following
Theorem 1.{\it If $|\xi_i-\mathbf E\xi_i|\le\nu$ with probability $1$ and for all $t\in R^k$
$$ \mathbf E(\xi_1+\dots+\xi_n-\mathbf E(\xi_1+\dots+\xi_n),t)^2\le(t,t), $$
then, for $\nu<1$,
$$ \mathfrak L_{\Pi}(F_n,G_n)\le c\nu\biggl(\ln\frac{1}{\nu}\biggr)^3 $$
where the constant $c$ depends on the dimension $k$ and on the choice of the norm $|\cdot|$ but not on characteristics of $F_n$ or $G_n$.}