Asymptotic expansion of Gaussian chaos via probabilistic approach

For a centered d -dimensional Gaussian random vector ξ = ( ξ 1 , … , ξ d ) and a homogeneous function h : ℝ d → ℝ we derive asymptotic expansions for the tail of the Gaussian chaos h ( ξ ) 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 ( ξ ) and its density at infinity and then discuss possible extensions for some general ξ with polar representation.