Asymptotic expansion of Gaussian chaos via probabilistic approach

Hashorva, Enkelejd and Korshunov, Dmitry and Piterbarg, Vladimir I. (2015) Asymptotic expansion of Gaussian chaos via probabilistic approach. Extremes, 18 (3). pp. 315-347. ISSN 1386-1999

Abstract

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.