Asymptotic expansion of Gaussian chaos via probabilistic approach

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.Keywords Wiener chaos · polynomial chaos · Gaussian chaos · multidimensional normal distribution · subexponential distribution · determinant of a random matrix · Gaussian orthogonal ensemble · diameter of random Gaussian clouds · max-domain of attraction Mathematics Subject Classification (2010) MSC 60J57 · MSC 60E05 · MSC 60E99

On the asymptotic Laplace method and its application to random chaos

“…Corollary 7 was proved in [17] (see also [18]) by a straightforward probabilistic method. References to questions pertaining to various Gaussian models can be found in [17].…”

Extremes of Gaussian chaos processes with trend

“…then we can choose W large enough such that Next, we give the theorem about asymptotic expansions for probability density and tail probability of the Gaussian random chaos in [18,21,22].…”

Large extremes of Gaussian chaos processes

“…Notice that the asymptotical behaviors of the prob ability density of g(ξ(0)) as u → ∞ and its tail distribu tion are evaluated in [1][2][3]. These results are used in the proof of the Theorem 1.…”

“…The starting point of our consideration is the theory of large deviations of Gaussian chaos developed in [1] and other publications of the authors [2,3], see the bibliography lists therein. The method of studying is a modification of the Double Sum Method, see [7,8].…”