On sharp large deviations for sums of random vectors and multidimensional Laplace approximation

Let $X, X_i,i\geq 1$, be a sequence of independent and identically distributed random vectors in $R^d$. Consider the partial sum $S_n:=X_1+\cdots +X_n$. Under some regularity conditions on the distribution of $X$, we obtain an asymptotic formula for $P\{S_n\in nA\}$, where $A$ is an arbitrary Borel set. Several corollaries follow, one of which asserts that, under the same regularity conditions, for any Borel set $A$, $\lim_{n\to\infty}n^{-1}\log P\{S_n\in nA\} =-I(A)$, where $I$ is a large deviation functional. We also prove a multidimensional Laplace-type approximation that allows an explicit calculation of the sharp large deviation probability typically when the set $A$ has a smooth boundary.