Dependent Random Fields Research Articles

Information model of economy

A new stochastic model of economy is developed that takes into account the choice of consumers are the de- pendent random fields. Axioms of such a model are formulated. The existence of random fields of consumer's choice and decision making by firms are proved. New notions of conditionally independent random fields and random fields of evaluation of information by consumers are introduced. Using the above mentioned random fields the random fields of consumer choice and decision makin g by firms are constructed. The theory of economic equilibrium is developed. The proposed stochastic model of economy contains a new approach to the description of consumer choice and making a decision by firm. This description is based on real observations of consumer choice that is described by probability measures ensemble given on budget sets of consumer. This description differs from classical description of consumer choice and making a decision by firm because consumers and firms make their choice and decision having information about the state of economy and their choice or decision depend on the available information. Our approach permits to construct random fields of consumer choice and making a decision by firms that are dependent random fields. This is very important because as Pareto showed the distribution of wealth for several nations has a power law. This result it is impossible to obtain if we restrict ourselves to classical description of consumer choice and take into account that consumers make their choice independently (6). We develop the theory of economic equilibrium that is as strict as in the classical approach (2). We also develop algorithms of finding equilibrium states that are constructive and are also applicable to the classical case (4,5). Such an approach solves the problem that the wealth in several societies is distributed according to Pareto law (7,6).

Strongly dependent Gaussian scenarios for the Burgers turbulence problem with quadratic external potential

We study the limiting distributions of a rescaling solution of the heat and Burgers equations, with quadratic external potential, in the case where the initial velocity potential is a strongly dependent Gaussian random field.

The invariance principle for a class of dependent random fields

Sufficient conditions for the tightness of a family of distributions of partial sum set-indexed processes constructed from symmetric random fields are obtained in this paper. We require that the moments of order s, s > 2, exist. The dependence structure of the field is described by the β1-mixing coefficients decreasing with a power rate. Assuming that a field is stationary and applying a result of D. Chen (1991) on the convergence of finite-dimensional distributions of the processes we obtain the invariance principle.

Random-field model for the elasticity tensor of anisotropic random media

This Note deals with the construction of a non-Gaussian positive definite matrix-valued random field whose mathematical properties allow the fourth-order elasticity tensor of random non homogeneous anisotropic three dimensional elastic media to be modelled. If the usual parametric probabilistic approach was used, then 21 mutually dependent random fields should be modelled and identified by using experimental data. Such an approach would be very difficult because the systems of the marginal probability distributions of these random fields have to be identified due to the fact that, for a boundary value problem, the displacement field of the random medium is a non-linear mapping of the random elasticity tensor. The theory presented in this paper allows such a probabilistic model of the fourth-order elasticity tensor field to be constructed and depends only of four scalar parameters: three spatial correlation lengths and one parameter allowing the level of the random fluctuations to be controlled. To cite this article: C. Soize, C. R. Mecanique 332 (2004).

Maximal Inequality for Weakly Dependent Random Fields

We obtain a maximal inequality for weakly dependent random fields associated with decreasing covariances of functions (of a certain class) of elements of the field as the distance between the indexing sets tends to infinity.

Convergence rates in the strong laws for a class of dependent random fields

By using a Rosenthal type inequality established in this paper, the complete convergence rates in the strong laws for a class of dependent random fields are discussed. And the result obtained extends those for ρ−-mixing random fields, ρ*-mixing random fields and negatively associated fields.

The Impact of Bootstrap Methods on Time Series Analysis

Sparked by Efron's seminal paper, the decade of the 1980s was a period of active research on bootstrap methods for independent data--mainly i.i.d. or regression set-ups. By contrast, in the 1990s much research was directed towards resampling dependent data, for example, time series and random fields. Consequently, the availability of valid nonparametric inference procedures based on resampling and/or subsampling has freed practitioners from the necessity of resorting to simplifying assumptions such as normality or linearity that may be misleading.

Limit theorems for the empirical distribution function in the spatial case

Functional central limit theorems are proved for the empirical distribution function of strictly stationary and weakly dependent random fields. The theorems cover the discrete and the continuous parameter fields and the case when the observations become dense in a sequence of increasing domains.

Rates in the Empirical Central Limit Theorem for Stationary Weakly Dependent Random Fields

A weak dependence condition is derived as the natural generalization to random fields on notions developed in Doukhan and Louhichi (1999). Examples of such weakly dependent fields are defined. In the context of a weak dependence coefficient series with arithmetic or geometric decay, we give explicit bounds in Prohorov metric for the convergence in the empirical central limit theorem. For random fields indexed by &Zopf d , in the geometric decay case, rates have the form n −1/(8d+24) L(n), where L(n) is a power of log(n).

Interacting diffusions in a random medium: comparison and longtime behavior

We consider a collection of linearly interacting diffusions (indexed by a countable space) in a random medium. The diffusion coefficients are the product of a space–time dependent random field (the random medium) and a function depending on the local state. The main focus of the present work is to establish a comparison technique for systems in the same medium but with different state dependence in the diffusion terms. The technique is applied to generalize statements on the longtime behavior, previously known only for special choices of the diffusion function. One of these special choices, which we employ as a reference model, is that of interacting Fisher–Wright diffusions in a catalytic medium where duality was used to obtain detailed results. The other choice is that of interacting Feller's branching diffusions in a catalytic medium which is itself an (autonomous) branching process and where infinite divisibility was used as the main tool.

On the connectivity properties of the complementary set in fractal percolation models

We study the connectivity properties of the complementary set in Poisson multiscale percolation model and in Mandelbrot's percolation model in arbitrary dimension. By using a result about majorizing dependent random fields by Bernoulli fields, we prove that if the selection parameter is less than certain critical value, then, by choosing the scaling parameter large enough, we can assure that there is no percolation in the complementary set.

Central Limit Theorems for Asymptotically Negatively Associated Random Fields

The aim of this paper is to investigate the central limit theorems for asymptotically negatively dependent random fields under lower moment conditions or the Lindeberg condition. Results obtained improve a central limit theorem of Roussas [11] for negatively assiated fields and the main results of Su and Chi [18], and also include a central limit of theorem for weakly negatively associated random variables similar to that of Burton et al. [20].

A Functional Central Limit Theorem for Asymptotically Negatively Dependent Random Fields

Let X k ; k ∈ N d be a random field which is asymptotically negative dependent in a certain sense. Define the partial sum process in the usual way so that \(W_n \left( t \right) = \sigma _n^{{\text{ - 1}}} \sum\nolimits_{m \leqq n \cdot t} {\left( {X_m - EX_m } \right)} \quad {\text{for}}\quad t \in \left[ {0,1} \right]^d\), where \(\sigma _n^{\text{2}} = {\text{Var}}\left( {\sum\nolimits_{m \leqq n} {X_m } } \right)\). Under some suitable conditions, we show that W n (·) converges in distribution to a Brownian sheet. Direct consequences of the result are functional central limit theorems for negative dependent random fields. The result is based on some general theorems concerning asymptotically negative dependent random fields, which are of independent interest.

On a Lower Bound of the Rate of Convergence in the Central Limit Theorem for m-Dependent Random Fields

We derive a lower bound of the rate of convergence in the central limit theorem for real m-dependent random fields under the finiteness of the fifth absolute moments of summands.

Geometry of \(\mathbb{Z}^d \) and the Central Limit Theorem for Weakly Dependent Random Fields

We study the asymptotic distribution of \(S_N (A,X) = \sqrt {(2N + 1)} ^{ - d} (\sum {_{n \in A_N } } X_n )\) where A is a subset of \(\mathbb{Z}^d \), AN= A∩[−N, N] d , v(A) = lim N card(AN) (2N+1) −d ∈(0, 1) and X is a stationary weakly dependent random field. We show that the geometry of A has a relevant influence on the problem. More specifically, SN(A, X) is asymptotically normal for each X that satisfies certain mixting hypotheses if and only if \(F_N (n;A) = {\text{card\{ }}A_N^c \cap {\text{(}}n + A_N {\text{)\} (}}2N + 1{\text{)}}^{ - d} \) has a limit F(n; A) as N → ∞ for each \(n \in \mathbb{Z}^d \). We also study the class of sets A that satisfy this condition.

Asymptotics for the local time of a strongly dependent vector-valued Gaussian random field

Asymptotics for the local time of a strongly dependent vector-valued Gaussian random field

Effect of vortex fluctuations on 205Tl spin-lattice relaxation in the mixed state of Tl2Ba2CuO6.

The spin-lattice relaxation rate W on $^{205}\mathrm{Tl}$ has been measured in the mixed state of a ${\mathrm{Tl}}_{2}$${\mathrm{Ba}}_{2}$${\mathrm{CuO}}_{6}$ aligned powder sample with ${\mathit{T}}_{\mathit{c}}$=100 K. At T25 K, W\ensuremath{\propto}T and decreases with increasing magnetic field. This behavior is explained as caused by fluctuations of pancake vortices; the latter induce a time dependent random magnetic field which causes the relaxation of nuclear spins.

A note on asymptotic expansions for sums over a weakly dependent random field with application to the Poisson and Strauss processes

Previous results on Edgeworth expansions for sums over a random field are extended to the case where the strong mixing coefficient depends not only on the distance between two sets of random variables, but also on the size of the two sets. The results are applied to the Poisson and the Strauss point processes, giving rise also to local limit results.

On the Large Deviation Principle for Stationary Weakly Dependent Random Fields

The large deviation principle for the empirical field of a stationary $\mathbb{Z}^d$-indexed random field is proved under strong mixing dependence assumptions. The strong mixing coefficients considered allow us to separate the ratio-mixing condition used in the literature into a part directly responsible for the (nonuniform) large deviation principle and another one, which is used when the state space is noncompact. Results are applied to obtain variants of recent large deviation theorems for Markov chains and for Gibbs fields. The proofs are based on a new criterion for the large deviation principle which is stated in Appendix C.

On the spectral density and asymptotic normality of weakly dependent random fields

For weakly stationary random fields, conditions on coefficients of “linear dependence” are given which are, respectively, sufficient for the existence of a continuous spectral density, and necessary and sufficient for the existence of a continuous positive spectral density. For strictly stationary random fields, central limit theorems are proved under the corresponding “unrestricted ϱ-mixing” condition and just finite or “barely infinite” second moments. No mixing rate is assumed.