Geodesic
On the estimate of the rate of convergence with respect to the system of balls in the central limit theorem in $R^k$
Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 2, pp. 440-445 Cet article a éte moissonné depuis la source Math-Net.Ru

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We obtain an estimate of the order $O(n^{-1/2)}$ for the rate of convergence with respect to the balls in the central limit theorem in $R^k$. The main part of the estimate does not depend on the dimensional parameter $k$. This estimate accounts the closeness of the distributions of summands to the normal distribution. 
@article{TVP_1983_28_2_a21,
     author = {V. V. Senatov},
     title = {On the estimate of the rate of convergence with respect to the system of balls in the central limit theorem in $R^k$},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {440--445},
     year = {1983},
     volume = {28},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1983_28_2_a21/}
}
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V. V. Senatov. On the estimate of the rate of convergence with respect to the system of balls in the central limit theorem in $R^k$. Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 2, pp. 440-445. http://geodesic.mathdoc.fr/item/TVP_1983_28_2_a21/