Geodesic
On the accuracy of Gaussian approximation for the probability of hitting a ball
Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 2, pp. 270-278 
V. V. Yurinskiǐ. On the accuracy of Gaussian approximation for the probability of hitting a ball. Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 2, pp. 270-278. http://geodesic.mathdoc.fr/item/TVP_1982_27_2_a5/
@article{TVP_1982_27_2_a5,
     author = {V. V. Yurinskiǐ},
     title = {On the accuracy of {Gaussian} approximation for the probability of hitting a~ball},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {270--278},
     year = {1982},
     volume = {27},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1982_27_2_a5/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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].