Evolution in a Gaussian Random Field

We consider an evolution process in a Gaussian random field V(q) with the mean ‹V(q)› = 0 and the correlation function W(|q – q|) ‹V(q)V(q)›, where q ^d and d is the dimension of the Euclidean space ^d . For the value ‹G(q,t;q ₀)›, t > 0, of the Green's function of the evolution equation averaged over all realizations of the random field, we use the Feynman–Kac formula to establish an integral equation that is invariant with respect to a continuous renormalization group. This invariance property allows using the renormalization group method to find an asymptotic expression for ‹G(q,t;q ₀)› as |q – q ₀| and t .     

On asymptotic properties of the empirical correlation matrix of a homogeneous vector-valued Gaussian field

We investigate properties of the empirical correlation matrix of a centered stationary Gaussian vector field in various function spaces. We prove that, under the condition of integrability of the square of the spectral density of the field, the normalization effect takes place for a correlogram and integral functional of it.     

Modeling Gaussian Isotropic Random Fields on a Sphere

We construct models of Gaussian isotropic random fields on a sphere in an n-dimensional space. The models approximate the fields with a given reliability and accuracy in the space L _p , p 2.     

Nonstationary Spatial Gaussian Markov Random Fields

Thin-plate splines have been widely used as spatial smoothers. In this article, we present a Bayesian adaptive thin-plate spline (BATS) suitable for modeling nonstationary spatial data. Fully Bayesian inference can be carried out through efficient Markov chain Monte Carlo simulation. Performance is demonstrated with simulation studies and by an application to a rainfall dataset. A FORTRAN program implementing the method, a proof of the theorem, and the dataset in this article are available online.     

Random matrices with complex Gaussian entries

In this paper we give new and purely analytical proofs of a number of classical results on the asymptotic behavior of large random matrices of complex Wigner type (the GUE-case) or of complex Wishart type: Wigner's semi-circle law, the Harer-Zagier recursion formula, the Marchenko-Pastur law, the Geman-Silverstein results on the largest and smallest eigenvalues and other related results. Our approach is based on the derivation of explicit formulae for the moment generating functions for random matrices of the two considered types.     

Looking at a Gaussian process through a window

We show that an observer recording only the presence or absence of a Gaussian process in a window can reconstruct its correlation function.     

On Gaussian Self-Similar Random Fields on a 2-Adic Plane

Russian Mathematics - The article describes a new type of Gaussian fields on a two-dimensional 2-adic space, invariant under the translation group and group of scaling transformations (self-similar...     

Discrimination with respect to a Gaussian process

Summary Let (N _t) and (Y _t), t in [0,1], be stochastic processes on (, , P). Suppose that (N _t) is Gaussian, m.s. continuous, zero mean, and vanishes a.s. at t=0. Let v _Y and v _N be the induced measures on ^[0,1]. Conditions are obtained for v _Y to be absolutely continuous w.r.t. v _N. Expressions for the Radon-Nikodym derivative are derived. Further results on these problems are obtained for measures induced on L ₂[0, 1] and on C[0, 1].Research supported by ONR Contracts N 00014-75-C-0491 and N 00014-81-K-0373     

Random walks on finite volume homogeneous spaces

Extending previous results by A. Eskin and G. Margulis, and answering their conjectures, we prove that a random walk on a finite volume homogeneous space is always recurrent as soon as the transition probability has finite exponential moments and its support generates a subgroup whose Zariski closure is semisimple.