Some Asymptotic Formulae for Gaussian Distributions
This paper considers asymptotic expansions of certain expectations which appear in the theory of large deviation for Gaussian random vectors with values in a separable real Hilbert space. A typical application is to calculation of the "tails" of distributions of smooth functionals,p(r)=P{[Phi](r-1[xi])[greater-or-equal, slanted]0},r-->[infinity], e.g., the probability that a centered Gaussian random vector hits the exterior of a large sphere surrounding the origin. The method provides asymptotic formulae for the probability itself and not for its logarithm in a situation, where it is natural to expect thatp(r)=c'rDÂ exp{-c''r2}. Calculations are based on a combination of the method of characteristic functionals with the Laplace method used to find asymptotics of integrals containing a fast decaying function with "small" support.
