A Geometric Approach to an Asymptotic Expansion for Large Deviation Probabilities of Gaussian Random Vectors

For the probabilities of large deviations of Gaussian random vectors an asymptotic expansion is derived. Based upon a geometric measure representation for the Gaussian law the interactions between global and local geometric properties both of the distribution and of the large deviation domain are studied. The advantage of the result is that the expansion coefficients can be obtained by making a series expansion of a surface integral avoiding the calculation of higher order derivatives.