A scalar valued random field {X (x)} x∈R d is called operator-scaling if for some d × d matrix E with positive real parts of the eigenvalues and some H > 0 we have= denotes equality of all finite-dimensional marginal distributions. We present a moving average and a harmonizable representation of stable operator scaling random fields by utilizing so called Ehomogeneous functions ϕ, satisfying ϕ(c E x) = cϕ(x). These fields also have stationary increments and are stochastically continuous. In the Gaussian case, critical Hölder-exponents and the Hausdorff-dimension of the sample paths are also obtained.

We investigate the sample paths regularity of operator scaling α-stable random fields. Such fields were introduced in [6] as anisotropic generalizations of self-similar fields and satisfy theIn the case of harmonizable operator scaling random fields, the sample paths are locally Hölderian and their Hölder regularity is characterized by the eigen decomposition of R d with respect to E. In particular, the directional Hölder regularity may vary and is given by the eigenvalues of E. In the case of moving average operator scaling random α-stable random fields, with α ∈ (0, 2) and d ≥ 2, the sample paths are almost surely discontinous.

Let X = {X(t), t ∈ R N } be a Gaussian random field with values in R d defined bywhere X 1 , . . . , X d are independent copies of a centered Gaussian random field X 0 . Under certain general conditions on X 0 , we study the hitting probabilities of X and determine the Hausdorff dimension of the inverse imageThe class of Gaussian random fields that satisfy our conditions includes not only fractional Brownian motion, the Brownian sheet, but also such anisotropic fields as fractional Brownian sheets, solutions to stochastic heat equation driven by space-time white noise and the operator-scaling Gaussian random fields with stationary increments constructed in [4].Running head: Inverse Images of Anisotropic Gaussian Random Fields 2000 AMS Classification numbers: 60G60; 60G15; 60G17; 28A80.

Abstract. We study generalized random fields which arise as rescaling limits of spatial configurations of uniformly scattered random balls as the mean radius of the balls tends to 0 or infinity. Assuming that the radius distribution has a power law behavior, we prove that the centered and re-normalized random balls field admits a limit with spatial dependence and self-similarity properties. In particular, our approach provides a unified framework to obtain all self-similar, translation and rotation invariant Gaussian fields. Under specific assumptions, we also get a Poisson type asymptotic field. In addition to investigating stationarity and self-similarity properties, we give L2-representations of the asymptotic generalized random fields viewed as continuous random linear functionals.

Abstract. In this paper, we propose a new and generic methodology for the analysis of texture anisotropy. The methodology is based on the stochastic modeling of textures by anisotropic fractional Brownian fields. It includes original statistical tests that permit to determine whether a texture is anisotropic or not. These tests are based on the estimation of directional parameters of the fields by generalized quadratic variations. Their construction is founded on a new theoretical result about the convergence of test statistics, which is proved in the paper. The methodology is applied to simulated data and discussed. We show that on a database composed of 116 full-field digital mammograms, about 60 percent of textures can be considered as anisotropic with a high level of confidence. These empirical results strongly suggest that anisotropic fractional Brownian fields are better-suited than the commonly used fractional Brownian fields to the modeling of mammogram textures.

Abstract. For a stationary random field (X j ) j∈Z d and some measure µ on R d , we consider the set-indexed weighted sum process Sn(A) = j∈Z d µ(nA ∩ R j ) 1 2 X j , where R j is the unit cube with lower corner j. We establish a general invariance principle under a p-stability assumption on the X j 's and an entropy condition on the class of sets A. The limit processes are selfsimilar set-indexed Gaussian processes with continuous sample paths. Using Chentsov's type representations to choose appropriate measures µ and particular sets A, we show that these limits can be Lévy (fractional) Brownian fields or (fractional) Brownian sheets.

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In this paper, we study modulus of continuity and rate of convergence of series of conditionally sub-Gaussian random fields. This framework includes both classical series representations of Gaussian fields and LePage series representations of stable fields. We enlighten their anisotropic properties by using an adapted quasi-metric instead of the classical Euclidean norm. We specify our assumptions in the case of shot noise series where arrival times of a Poisson process are involved. This allows us to state unified results for harmonizable (multi)operator scaling stable random fields through their LePage series representation, as well as to study sample path properties of their multistable analogous.