Asymptotic Expansion of Gaussian Chaos via Probabilistic Approach

Abstract

For a centered $d$-dimensional Gaussian random vector $\xi =({\xi }_1,\dots ,{\xi }_d)$ and a homogeneous function $h:R^d\to R$ we derive asymptotic expansions for the tail of the Gaussian chaos $h(\xi )$ given the function $h$ is sufficiently smooth. Three challenging instances of the Gaussian chaos are the determinant of a Gaussian matrix, the Gaussian orthogonal ensemble and the diameter of random Gaussian clouds. Using a direct probabilistic asymptotic method, we investigate both the asymptotic behaviour of the tail distribution of $h(\xi )$ and its density at infinity and then discuss possible extensions for some general $\xi$ with polar representation.

31 pages

References in corpus (3)

• Asymptotics of supremum distribution of a Gaussian process over a Weibullian time
• Moments of the Gaussian Chaos
• Extremes of Aggregated Dirichlet Risks

Cited by in corpus (4)

• Extremes of Aggregated Dirichlet Risks
• The diameter of a random elliptical cloud
• Gaussian Approximation of Perturbed Chi-Square Risks
• Extremal behavior of squared Bessel processes attracted by the Brown-Resnick process