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Estimation of anisotropic gaussian fields through Radon transform

Hermine Biermé, Frédéric Richard (2008)

ESAIM: Probability and Statistics

We estimate the anisotropic index of an anisotropic fractional brownian field. For all directions, we give a convergent estimator of the value of the anisotropic index in this direction, based on generalized quadratic variations. We also prove a central limit theorem. First we present a result of identification that relies on the asymptotic behavior of the spectral density of a process. Then, we define Radon transforms of the anisotropic fractional brownian field and prove that these processes admit...

Estimation of anisotropic Gaussian fields through Radon transform

Hermine Biermé, Frédéric Richard (2007)

ESAIM: Probability and Statistics

We estimate the anisotropic index of an anisotropic fractional Brownian field. For all directions, we give a convergent estimator of the value of the anisotropic index in this direction, based on generalized quadratic variations. We also prove a central limit theorem. First we present a result of identification that relies on the asymptotic behavior of the spectral density of a process. Then, we define Radon transforms of the anisotropic fractional Brownian field and prove that these processes...

Existence and asymptotic behaviour of some time-inhomogeneous diffusions

Mihai Gradinaru, Yoann Offret (2013)

Annales de l'I.H.P. Probabilités et statistiques

Let us consider a solution of a one-dimensional stochastic differential equation driven by a standard Brownian motion with time-inhomogeneous drift coefficient ρ sgn ( x ) | x | α / t β . This process can be viewed as a Brownian motion evolving in a potential, possibly singular, depending on time. We prove results on the existence and uniqueness of solution, study its asymptotic behaviour and made a precise description, in terms of parameters ρ , α and β , of the recurrence, transience and convergence. More precisely, asymptotic...

Existence et régularité höldérienne des fonctions de bosses

Moez Ben Abid (2009)

Colloquium Mathematicae

We discuss the almost sure existence of random functions that can be written as sums of elementary pulses. We then estimate their uniform Hölder regularity by applying some results on coverings by random intervals.

From almost sure local regularity to almost sure Hausdorff dimension for gaussian fields

Erick Herbin, Benjamin Arras, Geoffroy Barruel (2014)

ESAIM: Probability and Statistics

Fine regularity of stochastic processes is usually measured in a local way by local Hölder exponents and in a global way by fractal dimensions. In the case of multiparameter Gaussian random fields, Adler proved that these two concepts are connected under the assumption of increment stationarity property. The aim of this paper is to consider the case of Gaussian fields without any stationarity condition. More precisely, we prove that almost surely the Hausdorff dimensions of the range and the graph...

Images of Gaussian random fields: Salem sets and interior points

Narn-Rueih Shieh, Yimin Xiao (2006)

Studia Mathematica

Let X = X ( t ) , t ∈ ℝ N be a Gaussian random field in ℝ d with stationary increments. For any Borel set E ⊂ ℝ N , we provide sufficient conditions for the image X(E) to be a Salem set or to have interior points by studying the asymptotic properties of the Fourier transform of the occupation measure of X and the continuity of the local times of X on E, respectively. Our results extend and improve the previous theorems of Pitt [24] and Kahane [12,13] for fractional Brownian motion.

Incremental moments and Hölder exponents of multifractional multistable processes

Ronan Le Guével, Jacques Lévy Véhel (2013)

ESAIM: Probability and Statistics

Multistable processes, that is, processes which are, at each “time”, tangent to a stable process, but where the index of stability varies along the path, have been recently introduced as models for phenomena where the intensity of jumps is non constant. In this work, we give further results on (multifractional) multistable processes related to their local structure. We show that, under certain conditions, the incremental moments display a scaling behaviour, and that the pointwise Hölder exponent...

Joint continuity of the local times of fractional brownian sheets

Antoine Ayache, Dongsheng Wu, Yimin Xiao (2008)

Annales de l'I.H.P. Probabilités et statistiques

Let BH={BH(t), t∈ℝ+N} be an (N, d)-fractional brownian sheet with index H=(H1, …, HN)∈(0, 1)N defined by BH(t)=(BH1(t), …, BHd(t)) (t∈ℝ+N), where BH1, …, BHd are independent copies of a real-valued fractional brownian sheet B0H. We prove that if d<∑ℓ=1NHℓ−1, then the local times of BH are jointly continuous. This verifies a conjecture of Xiao and Zhang (Probab. Theory Related Fields124 (2002)). We also establish sharp local and global Hölder conditions for the local times of BH. These results...

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