Approximation of maximum of Gaussian random fields

This contribution is concerned with Gumbel limiting results for supremum ${\mathbb{M}}_{\mathit{n}}={\sup }_{\mathit{t}\in [\mathbf{0},{\mathit{T}}_{\mathit{n}}]}?|X_{\mathit{n}}(\mathit{t})|$ with $X_{\mathit{n}},\mathit{n}\in {\mathbb{N}}^2$ centered Gaussian random fields with continuous trajectories. We show first the convergence of a related point process to a Poisson point process thereby extending previous results obtained in [8] for Gaussian processes. Furthermore, we derive Gumbel limit results for ${\mathbb{M}}_{\mathit{n}}$ as $\mathit{n}\to \infty$ and show a second-order approximation for $\mathbb{E}{\{{\mathbb{M}}_{\mathit{n}}^p\}}^{1/p}$ for any $p\ge 1$.     

Markov approximation of homogeneous lattice random fields

We refine some well-known approximation theorems in the theory of homogeneous lattice random fields. In particular, we prove that every translation invariant Borel probability measure on the space X of finite-alphabet configurations on ^d, d1, can be weakly approximated by Markov measures _n with supp(_n)=X and with the entropies h(_n)h(). The proof is based on some facts of Thermodynamic Formalism; we also present an elementary constructive proof of a weaker version of this theorem.Mathematics Subject Classifications (2000): Primary 28D20, 37C85, 60G60; secondary 82B20Dedicated to Professor A. I. Vorobyov, member of the Russian Academy of Sciences and Director of the Hematology Research Center of the Russian Academy of Medical Sciences, on the occasion of his 75th birthday     

Scaling limits for point random fields

If X is a point random field on $\text{R}$^d then convergence in distribution of the renormalization C_λ|X_λ ? α_λ| as λ → ∞ to generalized random fields is examined, where C_λ > 0, α_λ are real numbers for λ > 0, and X_λ(f) = λ^?dX(f_λ) for $\text{f}{}_{\mathrm{\lambda }}\text{(x) = f(}\text{x}\text{λ}\text{)}$. If such a scaling limit exists then C_λ = λ^θg(λ), where g is a slowly varying function, and the scaling limit is self-similar with exponent θ. The classical case occurs when $\text{θ =}\text{d}\text{2}$ and the limit process is a Gaussian white noise. Scaling limits of subordinated Poisson (doubly stochastic) point random fields are calculated in terms of the scaling limit of the environment (driving random field). If the exponent of the scaling limit is $\text{θ =}\text{d}\text{2}$ then the limit is an independent sum of the scaling limit of the environment and a Gaussian white noise. If $\text{θ <}\text{d}\text{2}$ the scaling limit coincides with that of the environment while if $\text{θ >}\text{d}\text{2}$ the limit is Gaussian white noise. Analogous results are derived for cluster processes as well.     

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

The rank of sparse random matrices over finite fields

Let M be a random (n×n)-matrix over GF[q] such that for each entry M_ij in M and for each nonzero field element α the probability Pr[M_ij=α] is p/(q−1), where p=(log n−c)/n and c is an arbitrary but fixed positive constant. The probability for a matrix entry to be zero is 1−p. It is shown that the expected rank of M is n−𝒪(1). Furthermore, there is a constant A such that the probability that the rank is less than n−k is less than A/q^k. It is also shown that if c grows depending on n and is unbounded as n goes to infinity, then the expected difference between the rank of M and n is unbounded. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 10 , 407–419, 1997     

A class of small deviation theorems for functionals of random fields on a homogeneous tree

By constructing a non-negative martingale on a homogeneous tree, a class of small deviation theorems for functionals of random fields, the strong law of large numbers for the frequencies of occurrence of states and ordered couple of states for random fields, and the asymptotic equipartition property (AEP) for finite random fields are established. As corollary, the strong law of large numbers and the AEP for Markov chains indexed by a Cayley tree is obtained. Some known results are generalized in this paper.     

On weak convergence of random fields

We provide a characterization of compactness in the spaceD of functions of two variables defined on a unit square. The functions fromD have the property that their discontinuity points lie on smooth curves. Conditions for the tightness of probability measures inD and conditions for weak convergence of random fields with trajectories inD are derived. Vilnius Gediminas Technical University, Saulétekio 11; Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp 169–184, April–June, 1999. Translated by R. Banys     

A hyperfinite flat integral for generalized random fields

In this paper, a nonstandard construction of generalized white noise is established. This provides a (hyperfinite) flat integral representation of probability measures for generalized random fields derived as image probability measures of generalized white noise under certain measurable transformations, including Euclidean random fields obtained as convolution from generalized white noise with Euclidean kernels.     

A general rice formula,palm measures,and horizontal-window conditioning for random fields

In the first main result the mean (m?n)-dimensional Hausdorff measure of the set of crossing points of a level y ? $\text{R}$ⁿ by an m-dimensional continuous random vector field with values in R _ⁿ, m?n, is computed. The second one deals with horizontal-window conditional (Palm) distributions for such random fields. For this purpose, a general concept of Palm measures is introduced, which contains both the ‘stationary’ and the ‘nonstationary’ one.     

A posteriori error estimation for elliptic partial differential equations with small uncertainties

In this article, a finite element error analysis is performed on a class of linear and nonlinear elliptic problems with small uncertain input. Using a perturbation approach, the exact (random) solution is expanded up to a certain order with respect to a parameter that controls the amount of randomness in the input and discretized by finite elements. We start by studying a diffusion (linear) model problem with a random coefficient characterized via a finite number of random variables. The main focus of the article is the derivation of a priori and a posteriori error estimates of the error between the exact and approximate solution in various norms, including goal‐oriented error estimation. The analysis is then extended to a class of nonlinear problems. We finally illustrate the theoretical results through numerical examples, along with a comparison with the Stochastic Collocation method in terms of computational costs. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 175–212, 2016     

对一类随机算子方程的随机解的注记

给出在Banach空间中一类随机算子方程的随机解的某些新结果,它推广文[4]与文[6]中几个结果.     

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.     

A maximum a posteriori estimator for trajectories of diffusion processes

Let x _t be a diffusion process observed via a noisy sensor, whose output is y_t We consider the problem of evaluating the maximum a posteriori trajectory {x_s0≤ s ≤ t Based on results of Stratonovich [1] and Ikeda-Watanabe [2], we show that this estimator is given by the solution of an appropriate variational problem which is a slight modification of the "minimum energy" estimator. We compare our results to the non-linear filtering theory and show that for problems which possess a finite dimensional solution, our approach yields also explicit filters. For linear diffusions observed via linear sensors, these filters are identical to the Kalman-filter     

Non parametric estimation for random walks in random environment

We investigate the problem of estimating the cumulative distribution function (c.d.f.) $F$ of a distribution $\nu$ from the observation of one trajectory of the random walk in i.i.d. random environment with distribution $\nu$ on $\mathbb{Z}$. We first estimate the moments of $\nu$, then combine these moment estimators to obtain a collection of estimators ${\left({\stackrel{?}{F}}_n^M\right)}_{M\ge 1}$ of $F$, our final estimator is chosen among this collection by Goldenshluger–Lepski’s method. This estimator is easily computable. We derive convergence rates for this estimator depending on the Hölder regularity of $F$ and on the divergence rate of the walk. Our rate is minimal when the chain realizes a trade-off between a fast exploration of the sites, allowing to get more information and a larger number of visits of each site, allowing a better recovery of the environment itself.     

Parameter estimation of hidden periodic model in random fields

Two-dimensional hidden periodic model is an important model in random fields. The model is used in the field of two-dimensional signal processing, prediction and spectral analysis. A method of estimating the parameters for the model is designed. The strong consistency of the estimators is proved. Project supported by the National Natural Science Foundation of China (Grant No. 19571002).     

Testing the covariance function of stationary Gaussian random fields

In many problems, a specific function like h(⋅) is considered as the covariance function. Based on the asymptotic distribution of the periodogram and Euler characteristic, three methods are introduced to test the equality of the covariance function with h(⋅). Our analyses prove the accuracy of the power and scaling laws for the covariance function of metal surfaces.     

Efficient approximation of random fields for numerical applications

This article is dedicated to the rapid computation of separable expansions for the approximation of random fields. We consider approaches based on techniques from the approximation of non‐local operators on the one hand and based on the pivoted Cholesky decomposition on the other hand. Especially, we provide an a posteriori error estimate for the pivoted Cholesky decomposition in terms of the trace. Numerical examples are provided to validate and quantify the presented methods. Copyright © 2015 John Wiley & Sons, Ltd.     

Martingale-coboundary representation for a class of random fields

It is known that under some conditions, a stationary random sequence admits a representation as a sum of two sequences: one of them is a martingale difference sequence, and another one is a so-called coboundary. Such a representation can be used for proving some limit theorems by means of the martingale approximation. A multivariate version of such a decomposition is presented in the paper for a class of random fields generated by several commuting, noninvertible, probability preserving transformations In this representation, summands of mixed type appear, which behave with respect to some group of directions of the parameter space as reversed rnultiparameter martingale differences (in the sense of one of several known definitions), while they look as coboundaries relative to other directions. Applications to limit theorems will be published elsewhere. Bibliography: 14 titles.