On the accuracy of Gaussian approximation for the probability of hitting a ball

Let $X_1,X_2,\dots$ be independent random vectors in a separable Hilbert space $H$ such that $\mathbf EX_j=0$, $\mathbf E|X_j|^3\le L$ and $B$ is their common covariance operator. Let $Y$ be a centered Gaussian vector with a covariance operator $B/\operatorname{Sp} B$.
Theorem 1. {\it For $a\in H$, $r\ge 0$
$$ |\mathbf P\{|a+S_n|<r\}-\mathbf P\{|a+Y|<r\}|\le cL(\operatorname{Sp}B)^{-1/2}(1+|a|^3)n^{-1/2}, $$
where $S_n=(X_1+\dots+X_n)(n\operatorname{Sp}B)^{-1/2}$ and $c$ depends on the spectrum of $B/\operatorname{Sp}B$ only.}
The proof is based on the combination of results [2], [3].