Geodesic
An extension of the S.\,N.~Bernstein inequalities to multidimensional distributions
Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 2, pp. 266-274

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Let $X_1,\dots,X_n,\dots$ be a sequence of identically distributed independent random vectors in $R^m$ and $$ Y_n=\frac{X_1+\dots+X_n}{\sqrt n}, $$ Ir$\mathbf EX_j=0$, $|X_j|\le L$ and $n\ge m$, then $$ \mathbf P\{|Y_n|\ge r\}\le Ce^{-\frac{kr^2}{L^2}} $$ where $$ c\le1+\frac{e^{5/12}}{\pi/\sqrt2},\quad k\ge\frac1{8e^2}. $$ 
@article{TVP_1968_13_2_a3,
     author = {Yu. V. Prokhorov},
     title = {An extension of the {S.\,N.~Bernstein} inequalities to multidimensional distributions},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {266--274},
     publisher = {mathdoc},
     volume = {13},
     number = {2},
     year = {1968},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1968_13_2_a3/}
}
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Yu. V. Prokhorov. An extension of the S.\,N.~Bernstein inequalities to multidimensional distributions. Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 2, pp. 266-274. http://geodesic.mathdoc.fr/item/TVP_1968_13_2_a3/