Strong Markov Properties for Markov Random Fields

Markov properties and strong Markov properties for random fields are defined and discussed. Special attention is given to those defined by I. V. Evstigneev. The strong Markov nature of Markov random fields with respect to random domains such as [0, L], where L is a multidimensional extension of a stopping time, is explored. A special case of this extension is shown to generalize a result of Merzbach and Nualart for point processes. As an additional example, Evstigneev's Markov and strong Markov properties are considered for independent increment jump processes.     

Nonparametric Resampling for Homogeneous Strong Mixing Random Fields

Künsch (1989, Ann. Statist.17 1217-1241) and Liu ane Singh (1992, in Exploring Limits of Bootstrap (R. Le Page and L. Billard, Eds.), pp. 225-248, Wiley, New York) have recently introduced a block resampling method that is successful in deriving consistent bootstrap estimates of distribution and variance for the sample mean of a strong mixing sequence. Raïs and Moore (1990, in Interface ′90) and Raïs (1992, Ph.D. Thesis, University of Montreal) extended the results of Künsch and Liu and Singh in the case of the sample mean of a homogeneous strong mixing random field in two dimensions (n = 2). In this paper, the general case (n Z⁺) is considered, and a resampling technique for strong mixing random fields is formulated, which is an extension of the "blocks of blocks" resampling scheme for sequences in Politis and Romano (1992, Ann. Statist.20 (4) 1985-2007). The "blocks of blocks" method can be used to construct asymptotically correct confidence intervals for parameters of the whole (infinite-dimensional) joint distribution of the random field, for example, the spectral density at a point. A variation of the "blocks of blocks" resampling scheme that involves "wrapping" the data around on a torus will also be studied, in view of its property to yield an unbiased bootstrap distribution.     

Random Regularization of Brown Spectral Measure

We generalize a recent result of Haagerup; namely, we show that a convolution with a standard Gaussian random matrix regularizes the behavior of Fuglede-Kadison determinant and Brown spectral distribution measure. In this way, it is possible to establish a connection between the limit eigenvalues distributions of a wide class of random matrices and the Brown measure of the corresponding limits.     

Tangent Fields and the Local Structure of Random Fields

A tangent field of a random field X on ^N at a point z is defined to be the limit of a sequence of scaled enlargements of X about z. This paper develops general properties of tangent fields, emphasising their rich structure and strong invariance properties which place considerable constraints on their form. The theory is illustrated by a variety of examples, both of a smooth and fractal nature.     

Hausdorff Measure of Space Anisotropic Gaussian Processes with Non-stationary Increments

Let X={X（t）,t∈R₊}be a centered space anisotropic Gaussian process values in R^d with non-stationary increments,whose components are independent but may not be identically distributed.Under certain conditions,then almost surely c₁≤?-m(X（[0,1]）)≤c₂,where?denotes the exact Hausdorff measure associated with function■log log■for some 1≤k≤d,（α₁,···,α_d）∈(0,1]^].We also obtain the exact Hausdorff measure of the graph of X on[0,1].     

B值混合随机场的强大数律

本文讨论了取值于p型空间(1≤p＜2)上的混合随机场的强大数律和完全收敛性问题,获得了与独立同分布情形相类似的结果;改进了张立新(1998)的结果,并对该文中提出的一些问题作了回答.特别对ρ-混合,ρ*-混合实值随机序列部分和的大数定律给出了必要条件.     

Estimation of Parameters of Homogeneous Gaussian Random Fields

On the basis of the limit theorem for quadratic variation, we construct a consistent estimator for parameters of a homogeneous Gaussian random field of a certain class. We find confidence ellipsoids for estimators of this sort.     

Packing Dimension of Space-time Anisotropic Gaussian Random Fields

Acta Mathematica Sinica, English Series - Let X = {X(t) ∈ ℝd, t ∈ ℝN} be a centered space-time anisotropic Gaussian random field whose components are independent and satisfy...     

Wavelet Thresholding in Fixed Design Regression for Gaussian Random Fields

Journal of Fourier Analysis and Applications - In this paper, we consider block thresholding wavelet estimators of spatial regression functions on stationary Gaussian random fields observed over a...