Invariance principle for associated random fields

In 1984, C. M. Newman posed the problem of proving the invariance principle in distribution for associated random fields (i. e., fields satisfying the so-called FKG-inequalities)X={X_j, j∈Z^d} when d≥3. The solution of this problem for wide-sense stationary associated random fields is obtained here under slightly more restrictive conditions than those used by C. M. Newman and A. L. Wright for the strictly stationary case where d=1 and d=2. Partially supported by the Russian Foundation for Fundamental Research (grant No. 93-01-01454). Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part I.     

On the normal approximation for random fields via martingale methods

We prove a central limit theorem for strictly stationary random fields under a sharp projective condition. The assumption was introduced in the setting of random sequences by Maxwell and Woodroofe. Our approach is based on new results for triangular arrays of martingale differences, which have interest in themselves. We provide as applications new results for linear random fields and nonlinear random fields of Volterra-type.     

A central limit theorem for stationary random fields

Summary. We prove a central limit theorem for strictly stationary random fields under a projective assumption. Our criterion is similar to projective criteria for stationary sequences derived from Gordin's theorem about approximating martingales. However our approach is completely different, for we establish our result by adapting Lindeberg's method. The criterion that it provides is weaker than martingale-type conditions, and moreover we obtain as a straightforward consequence, central limit theorems for α-mixing or φ-mixing random fields. Received: 19 February 1997 / In revised form: 2 September 1997     

On the Almost Sure Central Limit Theorem for a Class of Z d -Actions

As an extension of earlier papers on stationary sequences, a concept of weak dependence for strictly stationary random fields is introduced in terms of so-called homoclinic transformations. Under assumptions made within the framework of this concept a form of the almost sure central limit theorem (ASCLT) is established for random fields arising from a class of algebraic Z ^d -actions on compact abelian groups. As an auxillary result, the central limit theorem is proved via Ch. Stein's method. The next stage of the proof includes some estimates which are specific for ASCLT. Both steps are based on making use of homoclinic transformations.     

Weak ergodicity and products of random matrices

A representation for a weakly ergodic sequence of (nonstochastic) matrices allows products of nonnegative matrices which eventually become strictly positive to be expressed via products of some associated stochastic matrices and ratios of values of a certain function. This formula used in a random setup leads to a representation for the logarithm of a random matrix product. If the sequence of random matrices is in addition stationary then automatically almost all sequences are weakly ergodic, and the representation is expressed in terms of an one-dimensional stationary process. This permits properties of products of random matrices to be deduced from the latter. Second moment assumptions guarantee that central limit theorems and laws of the iterated logarithm hold for the random matrix products if and only if they hold for the corresponding stationary process. Finally, a central limit theorem for some classes of weakly dependent stationary random matrices is derived doing away with the restriction of boundedness of the ratios of colum entries assumed by previous studies. Extensions beyond stationarity are discussed.     

Nonparametric regression on random fields with random design using wavelet method

We consider non-linear wavelet-based estimators of spatial regression functions with (known) random design on strictly stationary random fields, which are indexed by the integer lattice points in the \(N\)-dimensional Euclidean space and are assumed to satisfy some mixing conditions. We investigate their asymptotic rates of convergence based on thresholding of empirical wavelet coefficients and show that these estimators achieve nearly optimal convergence rates within a logarithmic term over a large range of Besov function classes \(B^{s}_{p,q}\). Therefore, wavelet estimators still achieve nearly optimal convergence rates for random fields and provide explicitly the extraordinary local adaptability.     

Spectral representation of intrinsically stationary fields

Transferring the concept of processes with weakly stationary increments to arbitrary locally compact Abelian groups two closely related notions arise: while intrinsically stationary random fields can be seen as a direct analog of intrinsic random functions of order k$k$ applied by G. Matheron in geostatistics, stationarizable random fields arise as a natural analog of definitizable functions in harmonic analysis. We concentrate on intrinsically stationary random fields related to finite-dimensional, translation-invariant function spaces, establish an orthogonal decomposition of random fields of this type, and present spectral representations for intrinsically stationary as well as stationarizable random fields using orthogonal vector measures.     

NA样本的递归密度估计

设$\{X_n,n\geq 1\}$是一个严平稳的负相协的随机变量序列, 其概率密度函数为$f(x)$.本文讨论了$f(x)$的递归核估计量的联合渐近正态性.     

ASYMPTOTIC NORMALITY OF KERNEL ESTIMATES OF A DENSITY FUNCTION UNDER ASSOCIATION DEPENDENCE

Let {X_n, n≥1} be a strictly stationary sequence of random variables, whichare either associated or negatively associated, f(·) be their common density. In this paper,the author shows a central limit theorem for a kernel estimate of f(·) under certain regularconditions.     

Singularity among selfsimilar Gaussian random fields with different scaling parameters and others

Abstract

It is shown in this paper that the probability measures generated by selfsimilar Gaussian random fields are mutually singular, whenever they have different scaling parameters. So are those generated from a selfsimilar Gaussian random field and a stationary Gaussian random field. Certain conditions are also given for the singularity of the probability measures generated from two Gaussian random fields whose covariance functions are Schoenberg–Lévy kernels, and for those from stationary Gaussian random fields with spectral densities.     

Models for Stationary Max-Stable Random Fields

Models for stationary max-stable random fields are revisited and illustrated by two-dimensional simulations. We introduce a new class of models, which are based on stationary Gaussian random fields, and whose realizations are not necessarily semi-continuous functions. The bivariate marginal distributions of these random fields can be calculated, and they form a new class of bivariate extreme value distributions.     

Finitary coding of Markov random fields

Summary There exists a finitary code from any stationary ergodic Markov random field to any i.i.d. random field of strictly lower entropy.     

Central limit theorems under weak dependence

This article is motivated by a central limit theorem of Ibragimov for strictly stationary random sequences satisfying a mixing condition based on maximal correlations. Here we show that the mixing condition can be weakened slightly, and construct a class of stationary random sequences covered by the new version of the theorem but not Ibragimov's original version. Ibragimov's theorem is also extended to triangular arrays of random variables, and this is applied to some kernel-type estimates of probability density.     

Stationary self-similar random fields on the integer lattice

We establish several methods for constructing stationary self-similar random fields (ssf's) on the integer lattice by “random wavelet expansion”, which stands for representation of random fields by sums of randomly scaled and translated functions, or more generally, by composites of random functionals and deterministic wavelet expansion. To construct ssf's on the integer lattice, random wavelet expansion is applied to the indicator functions of unit cubes at integer sites. We demonstrate how to construct Gaussian, symmetric stable, and Poisson ssf's by random wavelet expansion with mother wavelets having compact support or non-compact support. We also generalize ssf's to stationary random fields which are invariant under independent scaling along different coordinate axes. Finally, we investigate the construction of ssf's by combining wavelet expansion and multiple stochastic integrals.     

A functional central limit theorem for empirical processes under a strong mixing condition

This paper introduces a functional central limit theorem for empirical processes endowed with real values from a strictly stationary random field that satisfies an interlaced mixing condition. We proceed by using a common technique from Billingsley (Convergence of probability measures, Wiley, New York, 1999), by first obtaining the limit theorem for the case where the random variables of the strictly stationary ???-mixing random field are uniformly distributed on the interval [0, 1]. We then generalize the result to the case where the absolutely continuous marginal distribution function is not longer uniform. In this case we show that the empirical process endowed with values from the ???-mixing stationary random field, due to the strong mixing condition, doesn??t converge in distribution to a Brownian bridge, but to a continuous Gaussian process with mean zero and the covariance given by the limit of the covariance of the empirical process. The argument for the general case holds similarly by the application of a standard variant of a result of Billingsley (1999) for the space D(???, ??).     

Stationary and multi-self-similar random fields with stochastic volatility

This paper introduces stationary and multi-self-similar random fields which account for stochastic volatility and have type G marginal law. The stationary random fields are constructed using volatility modulated mixed moving average (MA) fields and their probabilistic properties are discussed. Also, two methods for parameterizing the weight functions in the MA representation are presented: one method is based on Fourier techniques and aims at reproducing a given correlation structure, the other method is based on ideas from stochastic partial differential equations. Moreover, using a generalized Lamperti transform we construct volatility modulated multi-self-similar random fields which have type G distribution.     

WEAK CONVERGENCE OF EMPIRICAL PROCESSES OF SEQUENCES OF STATIONARY RANDOM VARIABLES

In this paper, the author improves Yoshihara''s result(J.Multivariate Anal. 8(1978),584-588) and proves the weak convergence of empirical processes for sequence of p-mixing strictly stationary random variable with $\[\rho (n) = O({n^{ - \frac{1}{2} - \theta }}),\theta > 0\]$. Moreover, the author simplifies the complex proof of weak convergence of empirical processes wwith random index and gets the corresponding result for $\[\alpha \]$-mixing stationary random variables.     

Chung’s law of the iterated logarithm for anisotropic Gaussian random fields

In this paper we establish Chung’s law of the iterated logarithm for a class of anisotropic Gaussian random fields with stationary increments. This result is applicable to space–time Gaussian random fields and solution to the stochastic fractional heat equation.     

Convergence rates in the SLLN for some classes of dependent random fields

Let be a random field i.e. a family of random variables indexed by Nr, r?2. We discuss complete convergence and convergence rates under assumption on dependence structure of random fields in the case of nonidentical distributions. Results are obtained for negatively associated random fields, ρ^?-mixing random fields (having maximal coefficient of correlation strictly smaller then 1) and martingale random fields.     

强平稳LPQD序列更新过程的渐近正态性

本文讨论了强平稳LPQD随机变量列更新过程的渐近正态性问题.