Multidimensional integral limit theorems for large deviations

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
$$\mathbf P\{n^{-1/2}S_n\in D_n\},\qquad|a_n|\ge 1,\qquad|a_n|=o(\sqrt{n}),\qquad n\to\infty.$$