Asymptotic expansion of Gaussian chaos via probabilistic approach

• Published in: Extremes (Boston) Vol. 18; no. 3; pp. 315 - 347
• Main Authors: Hashorva, Enkelejd, Korshunov, Dmitry, Piterbarg, Vladimir I.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2015; Springer Nature B.V
• Subjects: Asymptotic expansions; Asymptotic methods; Asymptotic properties; Chaos theory; Civil Engineering; Density; Economics; Environmental Management; Finance; Gaussian; Hydrogeology; Insurance; Management; Mathematical analysis; Mathematics and Statistics; Normal distribution; Probabilistic methods; Probability theory; Quality Control; Random variables; Reliability; Safety and Risk; Statistics; Statistics for Business; Studies; Wiener chaos; Multidimensional normal distribution; Max-domain of attraction; Diameter of random Gaussian clouds; Gaussian orthogonal ensemble; Gaussian chaos; Subexponential distribution; Polynomial chaos; MSC 60E99; Determinant of a random matrix; MSC 60J57; MSC 60E05
• ISSN: 1386-1999; 1572-915X
• DOI: 10.1007/s10687-015-0215-3

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.