On the asymptotic Laplace method and its application to random chaos

Abstract

The asymptotics of the multidimensional Laplace integral for the case in which the phase attains its minimum on an arbitrary smooth manifold is studied. Applications to the study of the asymptotics of the distribution of Gaussian and Weibullian random chaoses are considered.

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.…”

“…We study the behavior of the probability for large u, that is, u → ∞, p > 0. By analogy with the notion of Gaussian chaos, see details in [2], we call the random process g(ξ(t)), the Gaussian chaos process. We assume here that the components ξ i (t), i = 1, 2, …, d, are independent stationary Gaussian processes with zero means, unit variances and identical covariance functions r(t).…”

“…We need notations and definitions from [2]. We have, g(x) = r β g(ϕ), with x = (r, ϕ), written in hyper spheric coordinates, r = |x|.…”

“…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].…”

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].…”