Dimension Results for Space-anisotropic Gaussian Random Fields

Let X = {X(t) ∈ R~d, t ∈ R~N} be a centered space-anisotropic Gaussian random field whose components satisfy some mild conditions. By introducing a new anisotropic metric in R~d, we obtain the Hausdorff and packing dimension in the new metric for the image of X. Moreover, the Hausdorff dimension in the new metric for the image of X has a uniform version.     

Hitting probabilities and the Hausdorff dimension of the inverse images of a class of anisotropic random fields

Let X = {X(t):t ∈ R~N} be an anisotropic random field with values in R~d.Under certain conditions on X,we establish upper and lower bounds on the hitting probabilities of X in terms of respectively Hausdorff measure and Bessel-Riesz capacity.We also obtain the Hausdorff dimension of its inverse image,and the Hausdorff and packing dimensions of its level sets.These results are applicable to non-linear solutions of stochastic heat equations driven by a white in time and spatially homogeneous Gaussian noise and anisotropic Guassian random fields.     

On intersections of independent anisotropic Gaussian random fields

Let XH = {XH(s),s ∈RN1} and X K = {XK(t),t ∈R N2} be two independent anisotropic Gaussian random fields with values in R d with indices H =(H1,...,HN1) ∈(0,1)N1,K =(K1,...,KN2) ∈(0,1) N2,respectively.Existence of intersections of the sample paths of X H and X K is studied.More generally,let E1■RN1,E2■RN2 and FRd be Borel sets.A necessary condition and a sufficient condition for P{(XH(E1)∩XK(E2))∩F≠Ф}>0 in terms of the Bessel-Riesz type capacity and Hausdorff measure of E1×E2×F in the metric space(RN1+N2+d,) are proved,where  is a metric defined in terms of H and K.These results are applicable to solutions of stochastic heat equations driven by space-time Gaussian noise and fractional Brownian sheets.     

Approximation of General Facets by Regular Facets with Respect to Anisotropic Total Variation Energies and Its Application to Crystalline Mean Curvature Flow

We show that every bounded subset of a euclidean space can be approximated by a set that admits a certain vector field, the so‐called Cahn‐Hoffman vector field, that is subordinate to a given anisotropic metric and has a square‐integrable divergence. More generally, we introduce a concept of facets as a kind of directed sets, and show that they can be approximated in a similar manner. We use this approximation to construct test functions necessary to prove the comparison principle for viscosity solutions of the level set formulation of the crystalline mean curvature flow that were recently introduced by the authors. As a consequence, we obtain the well‐posedness of the viscosity solutions in an arbitrary dimension, which extends the validity of the result in the previous paper.© 2018 Wiley Periodicals, Inc.     

Covariance tapering for prediction of large spatial data sets in transformed random fields

The best linear unbiased predictor (BLUP) is called a kriging predictor and has been widely used to interpolate a spatially correlated random process in scientific areas such as geostatistics. However, if an underlying random field is not Gaussian, the optimality of the BLUP in the mean squared error (MSE) sense is unclear because it is not always identical with the conditional expectation. Moreover, in many cases, data sets in spatial problems are often so large that a kriging predictor is impractically time-consuming. To reduce the computational complexity, covariance tapering has been developed for large spatial data sets. In this paper, we consider covariance tapering in a class of transformed Gaussian models for random fields and show that the BLUP using covariance tapering, the BLUP and the optimal predictor are asymptotically equivalent in the MSE sense if the underlying Gaussian random field has the Matérn covariance function.     

Kahane–Khintchine inequalities and functional central limit theorem for stationary random fields

We establish new Kahane–Khintchine inequalities in Orlicz spaces induced by exponential Young functions for stationary real random fields which are bounded or satisfy some finite exponential moment condition. Next, we give sufficient conditions for partial sum processes indexed by classes of sets satisfying some metric entropy condition to converge in distribution to a set-indexed Brownian motion. Moreover, the class of random fields that we study includes φ-mixing and martingale difference random fields.     

Example of a Gaussian Self-Similar Field With Stationary Rectangular Increments That Is Not a Fractional Brownian Sheet

We consider anisotropic self-similar random fields, in particular, the fractional Brownian sheet (fBs). This Gaussian field is an extension of fractional Brownian motion. It is well known that the fractional Brownian motion is a unique Gaussian self-similar process with stationary increments. The main result of this article is an example of a Gaussian self-similar field with stationary rectangular increments that is not an fBs. So we proved that the structure of self-similar Gaussian fields can be substantially more involved then the structure of self-similar Gaussian processes. In order to establish the main result, we prove some properties of covariance function for self-similar fields with rectangular increments. Also, using Lamperti transformation, we obtain properties of covariance function for the corresponding stationary fields.     

Simultaneous metric projection onto closed sets

We examine simultaneous metric projection by closed sets in a class of ordered normed spaces. First, we study simultaneous metric projection onto downward and upward sets and separation properties of these sets. The results obtained are used for examination of simultaneous metric projection by arbitrary closed sets, and we examine the minimization of the distance from a bounded set to an arbitrary closed set in a class of ordered normed spaces.     

Asymptotic behaviour of codes in rank metric over finite fields

In this paper, we recall some basic facts about the rank metric. We derive an asymptotic equivalent of the minimum rank distance of codes that reach the rank metric Gilbert–Varshamov bound. We show that the random codes reach GV-bound. Finally, we show that the optimal codes in rank metric have a packing density which is bounded by functions depending only on the base field and on the minimum distance. We show the potential interest in cryptographic applications.     

Estimation of intrinsic processes affected by additive fractal noise

Fractal Gaussian models have been widely used to represent the singular behavior of phenomena arising in different applied fields; for example, fractional Brownian motion and fractional Gaussian noise are considered as monofractal models in subsurface hydrology and geophysical studies Mandelbrot [The Fractal Geometry of Nature, Freeman Press, San Francisco, 1982 [13]]. In this paper, we address the problem of least-squares linear estimation of an intrinsic fractal input random field from the observation of an output random field affected by fractal noise (see Angulo et al. [Estimation and filtering of fractional generalised random fields, J. Austral. Math. Soc. A 69 (2000) 1-26 [2]], Ruiz-Medina et al. [Fractional generalized random fields on bounded domains, Stochastic Anal. Appl. 21 (2003a) 465-492], Ruiz-Medina et al. [Fractional-order regularization and wavelet approximation to the inverse estimation problem for random fields, J. Multivariate Anal. 85 (2003b) 192-216]. Conditions on the fractality order of the additive noise are studied to obtain a bounded inversion of the associated Wiener-Hopf equation. A stable solution is then obtained in terms of orthogonal bases of the reproducing kernel Hilbert spaces associated with the random fields involved. Such bases are constructed from orthonormal wavelet bases (see Angulo and Ruiz-Medina [Multiresolution approximation to the stochastic inverse problem, Adv. in Appl. Probab. 31 (1999) 1039-1057], Angulo et al. [Wavelet-based orthogonal expansions of fractional generalized random fields on bounded domains, Theoret. Probab. Math. Stat. (2004), in press]). A simulation study is carried out to illustrate the influence of the fractality orders of the output random field and the fractal additive noise on the stability of the solution derived.     

A solution to anisotropic suboptimal filtering problem by convex optimization

This paper considers a robust filtering problem for a linear discrete time invariant system with measured and estimated outputs. The system is exposed to random disturbances with imprecisely known distributions generated by an unknown stable shaping filter from the Gaussian white noise. The stochastic uncertainty of the input disturbance is measured by the mean anisotropy functional. The estimation error is quantified by the anisotropic norm which is a stochastic analogue of the H _∞ norm. A sufficient condition for an estimator to exist and ensure that the error is less than a given threshold value is derived in form of a convex inequality on the determinant of a positive definite matrix and two linear matrix inequalities. The suboptimal problem setting results to a set of the estimators ensuring the anisotropic norm of the error to be strictly bounded thereby providing some additional degree of freedom to impose some additional constraints on the estimator performance specification.     

On the Central Limit Theorem for Nonuniform φ-Mixing Random Fields

The partial-sum processes, indexed by sets, of a stationary nonuniform -mixing random field on the d-dimensional integer lattice are considered. A moment inequality is given from which the convergence of the finite-dimensional distributions to a Brownian motion on the Borel subsets of [0, 1] ^d is obtained. A Uniform CLT is proved for classes of sets with a metric entropy restriction and applied to certain Gibbs fields. This extends some results of Chen⁽⁵⁾ for rectangles. In this case and when the variables are bounded a simpler proof of the uniform CLT is given.     

An Analog of Titchmarsh’s Theorem for the Fourier–Walsh Transform

The space clos(X) of all nonempty closed subsets of an unbounded metric space X is considered. The space clos(X) is endowed with a metric in which a sequence of closed sets converges if and only if the distances from these sets to a fixed point θ are bounded and, for any r, the sequence of the unions of the given sets with the exterior balls of radius r centered at θ converges in the Hausdorff metric. The metric on clos(X) thus defined is not equivalent to the Hausdorff metric, whatever the initial metric space X. Conditions for a set to be closed, totally bounded, or compact in clos(X) are obtained; criteria for the bounded compactness and separability of clos(X) are given. The space of continuous maps from a compact space to clos(X) is considered; conditions for a set to be totally bounded in this space are found.     

The Structure of Extended Real-valued Metric Spaces

An extended metric on a set X is a distance function that satisfies the usual properties of a metric except that it can assume values of infinity, in addition to nonnegative real values. Given a metrizable space we exhibit a universal space for all extended metric spaces compatible with the topology. Defining a set in an extended metric space to be bounded if it is contained in a finite union of open balls, we characterize those bornologies on X that can be realized as bornologies of metrically bounded sets. We also consider a second possible definition of bounded set in this setting.     

Bornologies and metrically generated theories

Bornologies axiomatize an abstract notion of bounded sets and are introduced as collections of subsets satisfying a number of consistency properties. Bornological spaces form a topological construct, the morphisms of which are those functions which preserve bounded sets. A typical example is a bornology generated by a metric, i.e. the collection of all bounded sets for that metric. In a recent paper [E. Colebunders, R. Lowen, Metrically generated theories, Proc. Amer. Math. Soc. 133 (2005) 1547-1556] the authors noted that many examples are known of natural functors describing the transition from categories of metric spaces to the “metrizable” objects in some given topological construct such that, in some natural way, the metrizable objects generate the whole construct. These constructs can be axiomatically described and are called metrically generated. The construct of bornological spaces is not metrically generated, but an important large subconstruct is. We also encounter other important examples of metrically generated constructs, the constructs of Lipschitz spaces, of uniform spaces and of completely regular spaces. In this paper, the unified setting of metrically generated theories is used to study the functorial relationship between these constructs and the one of bornological spaces.     

On Global and Local Properties of the Trajectories of Gaussian Random Fields——A Look Through the Set of Limit Points

This paper studies the global and local properties of the trajectories of Gaussian random fields with stationary increments and proves sufficient conditions for Strassen's functional laws of the iterated logarithm at zero and infinity respectively.The sets of limit points of those Gaussian random fields are obtained.The main results are applied to fractional Riesz-Bessel processes and the sets of limit points of this field are obtained.     

关于随机解析算子函数的正族性质

该文在经典函数的正族理论基础上建立了随机解析算子函数的正族、一致有界和等度连续等概念，并在此意义下，给出了随机解析算子函数族内闭一致有界与等度连续、正族与一致有界的关系，以及随机解析算子函数族为正族的一个充分必要条件。     

A remark on random and equidistributed sequences

The average of the values of a function f on the points of an equidistributed sequence in [0, 1] ^s converges to the integral of f as soon as f is Riemann integrable. Some known low discrepancy sequences perform faster integration than independent random sampling (cf. [1]). We show that a small random absolutely continuous perturbation of an equidistributed sequence allows to integrate bounded Borel functions, and more generally that, if the law of the random perturbation doesn't charge polar sets, such perturbed sequences allow to integrate bounded quasi-continuous functions.     

Uniform dimension results for Gaussian random fields

Let X = {X(t), t ∈ ℝ ^N } be a Gaussian random field with values in ℝ ^d defined by

On Metric Boundedness Structures

Many years ago, S.-T. Hu gave necessary and sufficient conditions for a family of subsets of a metrizable space X to be the family of bounded sets for some admissible metric for the space. In this article, we show that in any noncompact metrizable space there are uncountably many distinct metric boundedness structures. Also, given an initial metric d for X, we look carefully at the problem of characterizing those boundedness structures determined by metrics uniformly equivalent to d. Applications to hyperspaces are given. Throughout, we rely on a dual approach to the study of metric boundedness.