Geodesic
Inequalities for the probabilities of large deviations in the multi-dimensional case
Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 4, pp. 755-759

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Let $X_1,\dots,X_n$ be independent random vectors with zero mean vectors. Let \begin{gather*} \Lambda_i=\mathbf E|X_i|^2,\quad M_i=\mathbf E|X_i|^3,\quad\Lambda=\frac1n\sum_{i=1}^n\Lambda_i,\quad M=\frac1n\sum_{i=1}^nM_i, \\ Y_n=\frac1{\sqrt n}(X_1+\dots+X_n) \end{gather*} We prove the following Theorem. There exist absolute constants $K_1$ and $K_2$ such that for any $x>0$ $$ \mathbf P(|Y_n|\ge x)\le4\exp(-K_1x^2/\Lambda)+K_2M/\sqrt nx^3 $$ 
@article{TVP_1971_16_4_a20,
     author = {\v{S}. S. \`Ebralidze},
     title = {Inequalities for the probabilities of large deviations in the multi-dimensional case},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {755--759},
     publisher = {mathdoc},
     volume = {16},
     number = {4},
     year = {1971},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1971_16_4_a20/}
}
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Š. S. Èbralidze. Inequalities for the probabilities of large deviations in the multi-dimensional case. Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 4, pp. 755-759. http://geodesic.mathdoc.fr/item/TVP_1971_16_4_a20/