Geodesic
On sharp large deviations for sums of random vectors and multidimensional Laplace approximation
Teoriâ veroâtnostej i ee primeneniâ, Tome 49 (2004) no. 4, pp. 743-774

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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.
Keywords: large deviations, exponential family, differential geometry of surfaces, asymptotic analysis, Laplace method
Mots-clés : Fourier transform. 
@article{TVP_2004_49_4_a6,
     author = {Ph. Barbe and M. Broniatowski},
     title = {On sharp large deviations for sums of random vectors and multidimensional {Laplace} approximation},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {743--774},
     publisher = {mathdoc},
     volume = {49},
     number = {4},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2004_49_4_a6/}
}
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Ph. Barbe; M. Broniatowski. On sharp large deviations for sums of random vectors and multidimensional Laplace approximation. Teoriâ veroâtnostej i ee primeneniâ, Tome 49 (2004) no. 4, pp. 743-774. http://geodesic.mathdoc.fr/item/TVP_2004_49_4_a6/