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Some Asymptotic Formulae for Gaussian Distributions

Author

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  • Yurinsky, V. V.

Abstract

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.

Suggested Citation

  • Yurinsky, V. V., 1996. "Some Asymptotic Formulae for Gaussian Distributions," Journal of Multivariate Analysis, Elsevier, vol. 56(2), pages 303-332, February.
    • Handle: RePEc:eee:jmvana:v:56:y:1996:i:2:p:303-332
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