Spectral analysis of some classes of multivariate random fields with isotropic property

In this paper we consider two classes of generalized random fields of second order on R^n with values in Hilbert space H with isotropic property: first - exponentially convex and isotropic random fields and second - homogeneous and isotropic random fields. The spectral representations for such fields and their covariance are obtained

On general strong laws of large numbers for fields of random variables

A general method to prove strong laws of large numbers for random fields is given. It is based on the Hájek-Rényi type method presented in Noszály and Tómács [14] and in Tómács and Líbor [16]. Noszály and Tómács [14] obtained a Hájek-Rényi type maximal inequality for random fields using moments inequalities. Recently, Tómács and Líbor [16] obtained a Hájek-Rényi type maximal inequality for random sequences based on probabilities, but not for random fields. In this paper we present a Hájek-Rényi type maximal inequality for random fields, using probabilities, which is an extension of the main results of Noszály and Tómács [14] by replacing moments by probabilities and a generalization of the main results of Tómács and Líbor [16] for random sequences to random fields. We apply our results to establishing a logarithmically weighted sums without moment assumptions and under general dependence conditions for random fields

Potential Vorticity Mixing in a Tangled Magnetic Field

A theory of potential vorticity (PV) mixing in a disordered (tangled) magnetic field is presented. The analysis is in the context of $\beta$-plane MHD, with a special focus on the physics of momentum transport in the stably stratified, quasi-2D solar tachocline. A physical picture of mean PV evolution by vorticity advection and tilting of magnetic fields is proposed. In the case of weak-field perturbations, quasi-linear theory predicts that the Reynolds and magnetic stresses balance as turbulence Alfv\'enizes for a larger mean magnetic field. Jet formation is explored quantitatively in the mean field-resistivity parameter space. However, since even a modest mean magnetic field leads to large magnetic perturbations for large magnetic Reynolds number, the physically relevant case is that of a strong but disordered field. We show that numerical calculations indicate that the Reynolds stress is modified well before Alfv\'enization -- i.e. before fluid and magnetic energies balance. To understand these trends, a double-average model of PV mixing in a stochastic magnetic field is developed. Calculations indicate that mean-square fields strongly modify Reynolds stress phase coherence and also induce a magnetic drag on zonal flows. The physics of transport reduction by tangled fields is elucidated and linked to the related quench of turbulent resistivity. We propose a physical picture of the system as a resisto-elastic medium threaded by a tangled magnetic network. Applications of the theory to momentum transport in the tachocline and other systems are discussed in detail.Comment: 17 pages, 10 figures, 2 table

Simulation of vibrations of a rectangular membrane with random initial conditions

A new method is proposed in this paper to construct models for solutions of boundary-value problems for hyperbolic equations with random initial conditions. We assume that the initial conditions are strictly sub-Gaussian random fields (in particular, Gaussian random fields with zero mean). The models approximate solutions with a given accuracy and reliability in the uniform metric

Scaling Limits in Models of Statistical Mechanics

The workshop brought together researchers interested in spatial random processes and their connection to statistical mechanics. The principal subjects of interest were scaling limits and, in general, limit laws for various two-dimensional critical models, percolation, random walks in random environment, polymer models, random fields and hierarchical diffusions. The workshop fostered interactions between groups of researchers in these areas and led to interesting and fruitful exchanges of ideas

A Strong Invariance Principle for Associated Random Fields

In this paper we generalize Yu’s strong invariance principle for associated sequences to the multi-parameter case, under the assumption that the covariance coefficient u(n) decays exponentially as n -> (infinity symbol). The main tools will be the Berkes-Morrow multi-parameter blocking technique, the Csörgö-Révész quantile transform method and the Bulinski rate of convergence in the CLT for associated random fields.strong invariance principle; associated random fields; blocking technique; quantile transform.

Loopy belief propagation and probabilistic image processing

Estimation of hyperparameters by maximization of the marginal likelihood in probabilistic image processing is investigated by using the cluster variation method. The algorithms are substantially equivalent to generalized loopy belief propagation

Automatic spectral density estimation for random fields on a lattice via bootstrap.

We consider the nonparametric estimation of spectral densities for secondorder stationary random fields on a d-dimensional lattice. We discuss some drawbacks of standard methods and propose modified estimator classes with improved bias convergence rate, emphasizing the use of kernel methods and the choice of an optimal smoothing number.We prove the uniform consistency and study the uniform asymptotic distribution when the optimal smoothing number is estimated from the sampled data.Spatial data; Spectral density; Smoothing number; Uniform asymptotic distribution; Bootstrap;

Mean anisotropy of homogeneous Gaussian random fields and anisotropic norms of linear translation-invariant operators on multidimensional integer lattices

Sensitivity of output of a linear operator to its input can be quantified in various ways. In Control Theory, the input is usually interpreted as disturbance and the output is to be minimized in some sense. In stochastic worst-case design settings, the disturbance is considered random with imprecisely known probability distribution. The prior set of probability measures can be chosen so as to quantify how far the disturbance deviates from the white-noise hypothesis of Linear Quadratic Gaussian control. Such deviation can be measured by the minimal Kullback-Leibler informational divergence from the Gaussian distributions with zero mean and scalar covariance matrices. The resulting anisotropy functional is defined for finite power random vectors. Originally, anisotropy was introduced for directionally generic random vectors as the relative entropy of the normalized vector with respect to the uniform distribution on the unit sphere. The associated a-anisotropic norm of a matrix is then its maximum root mean square or average energy gain with respect to finite power or directionally generic inputs whose anisotropy is bounded above by a >= 0. We give a systematic comparison of the anisotropy functionals and the associated norms. These are considered for unboundedly growing fragments of homogeneous Gaussian random fields on multidimensional integer lattice to yield mean anisotropy. Correspondingly, the anisotropic norms of finite matrices are extended to bounded linear translation invariant operators over such fields