Geodesic
Multidimensional integral limit theorems for large deviations
Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 1, pp. 62-82

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Let $S_n=X^{(1)}+\dots+X^{(n)}$ be a sum of independent identically distributed random vectors in $R^s$ and let $\{D_n\}$, $D_n\subset R^s$, be a sequence of convex Borel sets, for $n=1,2,\dots$. Let the point $a_n$ be the point of $D_n$ which is nearest to the origin. Under general conditions we obtain Cramer's type asymptotical formulas for $$ \mathbf P\{n^{-1/2}S_n\in D_n\},\qquad|a_n|\ge 1,\qquad|a_n|=o(\sqrt{n}),\qquad n\to\infty. $$ 
@article{TVP_1983_28_1_a3,
     author = {A. K. Ale\v{s}kevi\v{c}iene},
     title = {Multidimensional integral limit theorems for large deviations},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {62--82},
     publisher = {mathdoc},
     volume = {28},
     number = {1},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1983_28_1_a3/}
}
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A. K. Aleškevičiene. Multidimensional integral limit theorems for large deviations. Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 1, pp. 62-82. http://geodesic.mathdoc.fr/item/TVP_1983_28_1_a3/