Geodesic
Some inequalities for sums of multidimensional random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 1, pp. 155-159

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A sequence of $k$-dimensional independent identically distributed random variables $\xi_1,\xi_2,\dots$ is considered and probabilities $$ \mathbf P(s_n\in D_n), $$ where $s_n=\sum_{i=1}^n\xi_i$, $D_n$ is a sequence of regions of the $k$-dimensional space, are estimated. 
@article{TVP_1968_13_1_a12,
     author = {A. A. Borovkov},
     title = {Some inequalities for sums of multidimensional random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {155--159},
     publisher = {mathdoc},
     volume = {13},
     number = {1},
     year = {1968},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1968_13_1_a12/}
}
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A. A. Borovkov. Some inequalities for sums of multidimensional random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 1, pp. 155-159. http://geodesic.mathdoc.fr/item/TVP_1968_13_1_a12/