Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userMultiple Points of Gaussian Random Fields
101
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
This paper is concerned with the existence of multiple points of Gaussian random fields. Under the framework of Dalang et al. (2017), we prove that, for a wide class of Gaussian random fields, multiple points do not exist in critical dimensions. The result is applicable to fractional Brownian sheets and the solutions of systems of stochastic heat and wave equations.
rate research
Read More
We study the behaviour of the point process of critical points of isotropic stationary Gaussian fields. We compute the main term in the asymptotic expansion of the two-point correlation function near the diagonal. Our main result implies that for a generic field the critical points neither repel nor attract each other. Our analysis also allows to study how the short-range behaviour of critical points depends on their index.
Series expansions of isotropic Gaussian random fields on $mathbb{S}^2$ with independent Gaussian coefficients and localized basis functions are constructed. Such representations provide an alternative to the standard Karhunen-Lo`eve expansions of isotropic random fields in terms of spherical harmonics. Their multilevel localized structure of basis functions is especially useful in adaptive algorithms. The basis functions are obtained by applying the square root of the covariance operator to spherical needlets. Localization of the resulting covariance-dependent multilevel basis is shown under decay conditions on the angular power spectrum of the random field. In addition, numerical illustrations are given and an application to random elliptic PDEs on the sphere is analyzed.
We derive exact asymptotics of $$mathbb{P}left(sup_{tin mathcal{A}}X(t)>uright), ~text{as}~ utoinfty,$$ for a centered Gaussian field $X(t),~tin mathcal{A}subsetmathbb{R}^n$, $n>1$ with continuous sample paths a.s. and general dependence structure, for which $arg max_{tin {mathcal{A}}} Var(X(t))$ is a Jordan set with finite and positive Lebesque measure of dimension $kleq n$. Our findings are applied to deriving the asymptotics of tail probabilities related to performance tables and dependent chi processes.
We study the total mass of high points in a random model for the Riemann-Zeta function. We consider the same model as in [8], [2], and build on the convergence to Gaussian multiplicative chaos proved in [14]. We show that the total mass of points which are a linear order below the maximum divided by their expectation converges almost surely to the Gaussian multiplicative chaos of the approximating Gaussian process times a random function. We use the second moment method together with a branching approximation to establish this convergence.
The tube method or the volume-of-tube method approximates the tail probability of the maximum of a smooth Gaussian random field with zero mean and unit variance. This method evaluates the volume of a spherical tube about the index set, and then transforms it to the tail probability. In this study, we generalize the tube method to a case in which the variance is not constant. We provide the volume formula for a spherical tube with a non-constant radius in terms of curvature tensors, and the tail probability formula of the maximum of a Gaussian random field with inhomogeneous variance, as well as its Laplace approximation. In particular, the critical radius of the tube is generalized for evaluation of the asymptotic approximation error. As an example, we discuss the approximation of the largest eigenvalue distribution of the Wishart matrix with a non-identity matrix parameter. The Bonferroni method is the tube method when the index set is a finite set. We provide the formula for the asymptotic approximation error for the Bonferroni method when the variance is not constant.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
