Homogeneous Gaussian Fields Research Articles

Extremes of Homogeneous Two-Parametric Gaussian Fields at Discretization of Parameters

Extremes of Homogeneous Two-Parametric Gaussian Fields at Discretization of Parameters

The Persistence Exponents of Gaussian Random Fields Connected by the Lamperti Transform

The (fractional) Brownian sheet is the simplest example of a Gaussian random field $$X$$ whose covariance is the tensor product of a finite number (d) of nonnegative correlation functions of self- similar Gaussian processes. We consider the homogeneous Gaussian field $$Y$$ obtained by applying to X the exponential change of time (more precisely, the Lamperti transform). Under some assumptions, we prove the existence of the persistence exponents for both fields, $$X$$ and $$Y$$ , and find the relation between them. The exponent for any random function $$Z$$ is $$\psi (T)$$ if lim $$\ln P(Z({\mathbf{t}}) \le ,{\mathbf{t}} \in TG)/\psi (T) = - 1$$ , $$T > > {1}$$ , where $$G$$ is a d-dimensional region containing 0, and $$T$$ is a similarity coefficient. The considered problem was raised by Li and Shao [Ann. Probab. 32:1, 2004] and originally concerned the Brownian sheet.

Extremes of Homogeneous Gaussian Random Fields

Let {X(s, t): s, t ≥ 0} be a centred homogeneous Gaussian field with almost surely continuous sample paths and correlation function r(s, t) = cov(X(s, t), X(0, 0)) such that r(s, t) = 1 - |s|α1 - |t|α2 + o(|s|α1 + |t|α2 ), s, t → 0, with α1, α2 ∈ (0, 2], and r(s, t) < 1 for (s, t) ≠ (0, 0). In this contribution we derive an asymptotic expansion (as u → ∞) of P(sup(sn 1(u),tn 2(u)) ∈[0,x]∙[0,y] X(s, t) ≤ u), where n 1(u)n 2(u) = u 2/α1+2/α2 Ψ(u), which holds uniformly for (x, y) ∈ [A, B]2 with A, B two positive constants and Ψ the survival function of an N(0, 1) random variable. We apply our findings to the analysis of extremes of homogeneous Gaussian fields over more complex parameter sets and a ball of random radius. Additionally, we determine the extremal index of the discretised random field determined by X(s, t).

Extremes of Homogeneous Gaussian Random Fields

Let {X(s, t): s, t ≥ 0} be a centred homogeneous Gaussian field with almost surely continuous sample paths and correlation function r(s, t) = cov(X(s, t), X(0, 0)) such that r(s, t) = 1 - |s|α1 - |t|α2 + o(|s|α1 + |t|α2), s, t → 0, with α1, α2 ∈ (0, 2], and r(s, t) < 1 for (s, t) ≠ (0, 0). In this contribution we derive an asymptotic expansion (as u → ∞) of P(sup(sn1(u),tn2(u)) ∈[0,x]∙[0,y]X(s, t) ≤ u), where n1(u)n2(u) = u2/α1+2/α2Ψ(u), which holds uniformly for (x, y) ∈ [A, B]2 with A, B two positive constants and Ψ the survival function of an N(0, 1) random variable. We apply our findings to the analysis of extremes of homogeneous Gaussian fields over more complex parameter sets and a ball of random radius. Additionally, we determine the extremal index of the discretised random field determined by X(s, t).

Locally Homogeneous and Isotropic Gaussian Fields in Kriging

The paper deals with the application of the theory of locally homogeneous and isotropic Gaussian fields (LHIGF) to probabilistic modelling of multivariate data structures. An asymptotic model is al...

Locally Homogeneous and Isotropic Gaussian Fields in Kriging

The paper deals with the application of the theory of locally homogeneous and isotropic Gaussian fields (LHIGF) to probabilistic modelling of multivariate data structures. An asymptotic model is also studied, when the correlation function parameter of the Gaussian field tends to infinity. The kriging procedure is developed which presents a simple extrapolator by means of a matrix of degrees of the distances between pairs of the points of measurement. The resulting model is rather simple and can be defined only by the mean and variance parameters, efficiently evaluated by maximal likelihood method. The results of application of the extrapolation method developed for two analytically computed surfaces and estimation of the position of the spacecraft re-entering the atmosphere are given.

Lipschitz-Continuity of the Integrated Density of States for Gaussian Random Potentials

The integrated density of states of a Schroedinger operator with random potential given by a homogeneous Gaussian field whose covariance function is continuous, compactly supported and has positive mean, is locally uniformly Lipschitz-continuous. This is proven using a Wegner estimate.

Log-likelihood ratio test for detecting transient change

The limit behaviour of scan statistics in a form of maxima of moving sum processes with changing window size for detecting a transient change is studied. Approximate critical values obtained from tail behaviour of the limit variables that are maxima of some locally homogeneous Gaussian fields are compared with critical values obtained by simulation.

Random process in a homogeneous Gaussian field

We consider a random process in a spatial-temporal homogeneous Gaussian field V (q, t) with the mean EV = 0 and the correlation function W(|q−q′|, |t − t′|) ≡ E[V (q, t)V (q′, t′)], where \( \bold{q} \in {\mathbb{R}^d} \), \( t \in {\mathbb{R}^{+} } \), and d is the dimension of the Euclidean space \( {\mathbb{R}^d} \). For a “density” G(r, t) of the familiar model of a physical system averaged over all realizations of the random field V, we establish an integral equation that has the form of the Dyson equation. The invariance of the equation under the continuous renormalization group allows using the renormalization group method to find an asymptotic expression for G(r, t) as r → ∞ and t → ∞.

The generation of 2-D random velocity fields of groundwater flows via stream functions

This paper is a further development of former papers of the authors on the direct simulation of random velocity fields describing groundwater flow in aquifers of spatially variable, but stochastically homogeneous hydraulic conductivity. We use the so-called dual description of groundwater flow by means of stream functions. Further assumptions are therefore steady state flow, far boundaries and no sources and sinks. By a perturbation approach we get a first-order approximation of the spectra of the random stream function and the random velocity field. The generation of random velocity fields via stream functions is attractive because of its parsimony in the use of parameters. Only a scalar random field must be generated. Again we adopt Mikhailov's method of “partitioning and randomizing the spectrum” for the simulation of a homogeneous Gaussian field. We also tackle the problem of conditional simulation of random velocity fields honoring measured velocity values at several locations. The generation method is illustrated by a comparison of empirical and theoretical spectra.

Scalar statistics on the sphere: application to the cosmic microwave background

A method to compute several scalar quantities of cosmic microwave background (CMB) maps on the sphere is presented. We consider here four type of scalars: the Hessian matrix scalars, the distortion scalars, the gradient-related scalars and the curvature scalars. Such quantities are obtained directly from the spherical harmonic coefficients a lm of the map. We also study the probability density function of these quantities for the case of a homogeneous and isotropic Gaussian field, which are functions of the power spectrum of the initial field. From these scalars it is possible to construct a new set of scalars which are independent of the power spectrum of the field. We test our results using simulations and find good agreement between the theoretical probability density functions and those obtained from simulations. Therefore, these quantities are proposed to investigate the presence of non-Gaussian features in CMB maps. Finally, we show how to compute the scalars in the presence of anisotropic noise and realistic masks.

Sharp Gaussian regularity on the circle, and applications to the fractional stochastic heat equation

A sharp regularity theory is established for homogeneous Gaussian fields on the unit circle. Two types of characterizations for such a field to have a given almost-sure uniform modulus of continuity are established in a general setting. The first characterization relates the modulus to the field's canonical metric; the full force of Fernique's zero-one laws and Talagrand's theory of majorizing measures is required. The second characterization ties the modulus to the field's random Fourier series representation. As an application, it is shown that the fractional stochastic heat equation has, up to a non-random constant, a given spatial modulus of continuity if and only if the same property holds for a fractional antiderivative of the equation's additive noise; a random Fourier series characterization is also given.

Algorithm for Generating Samples of Homogeneous Gaussian Fields

A new algorithm is developed for generating samples of stationary Gaussian random fields. The algorithm is based on a model derived from the spectral representation theorem for weakly stationary random fields. The model consists of a superposition of a random number of waves with random amplitude and frequency, can match the second moment properties of any target random field, and becomes Gaussian as the intensity of two independent Poisson processes, defining the number of waves and their frequencies, increases indefinitely. In contrast to the current Monte Carlo simulation algorithms, the proposed algorithm: (1) does not produce periodic samples; and (2) does not require the discretization of the frequency domain. The proposed Monte Carlo algorithm is applied to generate samples of a stationary Gaussian field defined on ℝ2.

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.

Non-Gaussian along-wind response analysis in time and frequency domains

With special reference to the role of the non-Gaussian nature of the response, the procedures for along-wind frequency-domain analysis of MDOF structures are discussed. Assuming turbulence as a homogeneous Gaussian field, the dynamic response of structural systems is evaluated by using the power spectral density of the wind force in the case of the standard Gaussian analysis and by using both the power spectral density and bi-spectrum in the case of non-Gaussian analysis. The statistics of the response is used to estimate the peak value distribution. A parametric analysis, carried out on a steel lattice broadcasting antenna, and the comparison with suited time domain analyses of the response allowed to evaluate the reliability of the spectral methods.

Transport in multiscale log conductivity fields with truncated power variograms

Di Federico and Neuman [this issue] investigated mean uniform steady state groundwater flow in an unbounded domain with log hydraulic conductivity that forms a truncated multiscale hierarchy of statistically homogeneous and isotropic Gaussian fields, each associated with an exponential autocovariance. Here we present leading‐order expressions for the displacement covariance, and dispersion coefficient, of an ensemble of solute particles advected through such a flow field in two or three dimensions. Both quantities are functions of the mean travel distance s, the Hurst coefficient H, and the low‐ and high‐frequency cutoff integral scales λl and λu. The latter two are related to the length scales of the sampling window (region under investigation) and sample volume (data support), respectively. If one considers transport to be affected by a finite domain much larger than the mean travel distance, so that s ≪ λl < ∞, then an early preasymptotic regime develops during which longitudinal and transverse dispersivities grow linearly with s. If one considers transport to be affected by a domain which increases in proportion to s, then λl and s are of similar order and a preasymptotic regime never develops. Instead, transport occurs under a regime that is perpetually close to asymptotic under the control of an evolving scale λl ∼ s. We show that if, additionally, λu ≪ λl, then the corresponding longitudinal dispersivity grows in proportion to λl1+2H or, equivalently, s1+2H. Both these preasymptotic and asymptotic theoretical growth rates are shown to be consistent with the observed variation of apparent longitudinal Fickian dispersivities with scale. We conclude by investigating the effect of variable separations between cutoff scales on dispersion.

NON-GAUSSIAN RESPONSE OF MDOF WIND-EXPOSED STRUCTURES : ANALYSIS BY BICORRELATION FUNCTION AND BISPECTRUM

Abstract. The influence of non-Gaussian modeling of the wind action on the response of MDOF structures is considered. Assuming the turbulence as a homogeneous Gaussian field, the wind action field is described by means of the correlation and bicorrelation functions. Once known the spectrum and the bispectrum (by Fourier transform), the same quantities of the structural response are obtained by frequency domain analysis. This allows to evaluate the statistic moments of the response until the third moment and to estimate the peak value distribution. The procedure is applied to analyze a broadcasting antenna. Varying the structural damping and the wind characteristics, a comparison between Gaussian and non-Gaussian modeling is performed.

Reaction front propagation in a turbulent flow

The large-scale dynamics of a reaction front in a turbulent flow in the limit of large Reynolds number has been studied starting from the Kolmogorov-Petrovskii-Piskunov equation, modified by the random convection term. Random velocity has been assumed to be a homogeneous Gaussian field with Kolmogorov energy spectrum and infrared divergence. An upper bound for the position and speed of the reaction front in the long-time, large-distance limit has been derived by the method of random characteristics and a renormalization procedure. It has been shown that the infrared divergence of the random velocity field leads to the acceleration of a coarse-grained reaction front.

Conditions for Gaussian homogeneous fields to belong to classes

This paper is the continuation of [1], [2]. In it questions of approximation of functions are discussed, when a realization of the homogeneous Gaussian field belongs to the class H rp (G)0.

Conditions for the Equivalence of Distributions of Homogeneous Gaussian Fields

Previous article Next article Conditions for the Equivalence of Distributions of Homogeneous Gaussian FieldsS. M. KrasnitskiiS. M. Krasnitskiihttps://doi.org/10.1137/1134092PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] J. Feldman, Equivalence and perpendicularity of Gaussian processes, Pacif. J. Math., 8 (1958), 698–708, 9 (1959), pp. 1295–1296 CrossrefGoogle Scholar[2] Yu. A. Rozanov, Infinite-dimensional Gaussian distributions, Trudy Mat. Inst. Steklov., 108 (1968), 136 pp. (errata insert), (In Russian.) 45:7801 Google Scholar[3] A. M. Yaglom, Some classes of random fields in n-dimensional space related to stationary random processes, Theory Probab. Appl.,, 2 (1957), 273–320 LinkGoogle Scholar[4] I. M. Gel'fand and , N. Ya. Vilenkin, Generalized Functions, Vol. 1, Academic Press, New York, 1964 Google Scholar[5] I. A. Ibragimov and , Yu. A. Rozanov, Gaussian random processes, Applications of Mathematics, Vol. 9, Springer-Verlag, New York, 1978x+275, New York 80f:60038 CrossrefGoogle Scholar[6] Z. S. Zerakidze, On the equivalence of distributions of homogeneous fields, Trudy in-ta prikladnoi matematiki Tbilisskogo universiteta, 2 (1969), 215–220, (In Russian.) Google Scholar[7] A. V. Skorokhod and , M. I. Yadrenko, On absolute continuity of measures corresponding to homogeneous random fields, Theory Probab. Appl., 18 (1973), 27–40 10.1137/1118002 0282.60026 LinkGoogle Scholar[8] S. M. Krasnitskii, On conditions of equivalence and perpendicularity of measures corresponding to homogeneous Gaussian fields, Theory Probab. Appl., 18 (1973), 588–592 10.1137/1118075 0334.60021 LinkGoogle Scholar[9] V. S. Vladimirov, Generalized Functions in Mathematical Physics, Nauka, Moscow, 1976, (In Russian.) Google Scholar[10] I. M. Gel'fand and , G. E. Shilov, Generalized functions. Vol. 2. Spaces of fundamental and generalized functions, Translated from the Russian by Morris D. Friedman, Amiel Feinstein and Christian P. Peltzer, Academic Press, New York, 1968x+261, London 37:5693 Google Scholar[11] S. Bochner and , W. Martin, Several Complex Variables, Princeton Mathematical Series, vol. 10, Princeton University Press, Princeton, N. J., 1948ix+216 10,366a Google Scholar[12] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969xiv+676 41:1976 Google Scholar[13] S. M. Nikol'skii, Approximation of functions of several variables and imbedding theorems, Springer-Verlag, New York, 1975viii+418 51:11073 CrossrefGoogle Scholar[14] L. Hormander, Linear partial differential operators, Die Grundlehren der mathematischen Wissenschaften, Bd. 116, Academic Press Inc., Publishers, New York, 1963vii+287 28:4221 CrossrefGoogle Scholar[15] S. M. Krasnitskii, On conditions of equivalence of distributions of homogeneous Gaussian fields, Summaries of Reports of the 4th International Vilna conference on Probability Theory and Mathematical Statistics, Vol. 2, Institute of Math. and Cybern. of the Lithuanian Acad. Sci., 1985, 72–74, (In Russian.) Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Necessary and sufficient conditions for asymptotically optimal linear prediction of random fields on compact metric spacesThe Annals of Statistics, Vol. 50, No. 2 Cross Ref Volume 34, Issue 4| 1990Theory of Probability & Its Applications History Submitted:30 July 1986Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1134092Article page range:pp. 720-725ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics