Geodesic
Some remarks on multidimensional inegualities of the Bernstein--Kolmogorov type
Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 2, pp. 289-294

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Let $X_1,\dots,X_n$ be independent random vectors in $R^m$ for which $\mathbf EX_i=0$ and $Y=X_1+\dots+X_n$. In the paper upper bounds of the type of the Bernstein–Kolmogorov inequalities are obtained for the probabilities $\mathbf P(|Y|\ge t)$ in case when the components of $X_i$'s form a Lévy martingale (in the sense of definition (3)) or when these vectors have spherical distributions. The orders of magnitude of the estimates obtained can not be improved. 
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     author = {V. M. Zolotarev},
     title = {Some remarks on multidimensional inegualities of the {Bernstein--Kolmogorov} type},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {289--294},
     publisher = {mathdoc},
     volume = {13},
     number = {2},
     year = {1968},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1968_13_2_a5/}
}
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V. M. Zolotarev. Some remarks on multidimensional inegualities of the Bernstein--Kolmogorov type. Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 2, pp. 289-294. http://geodesic.mathdoc.fr/item/TVP_1968_13_2_a5/