1. Packing dimension results for anisotropic Gaussian random fields
- Author
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Dongsheng Wu, Yimin Xiao, Anne Estrade, Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematical Sciences, University of Alabama [Tuscaloosa] (UA), Department of Statistics and Probability (Michigan State University), Michigan State University [East Lansing], Michigan State University System-Michigan State University System, ANR-09-BLAN-0029,MATAIM,Modélisation de l'Anisotropie de Textures. Applications à l'Imagerie Médicale(2009), Mathématiques Appliquées à Paris 5 ( MAP5 - UMR 8145 ), Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National des Sciences Mathématiques et de leurs Interactions-Centre National de la Recherche Scientifique ( CNRS ), University of Alabama [Tuscaloosa] ( UA ), MATAIM, ANR-09-BLAN-029,MATAIM, ANR-09-BLAN-029, ANR-09-BLAN-0029,MATAIM,Modélisation de l'Anisotropie de Textures. Applications à l'Imagerie Médicale.(2009), Estrade, Anne, and Blanc - Modélisation de l'Anisotropie de Textures. Applications à l'Imagerie Médicale. - - MATAIM2009 - ANR-09-BLAN-0029 - Blanc - VALID
- Subjects
Statistics and Probability ,[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,Gaussian ,Hausdorff dimension ,01 natural sciences ,Gaussian random field ,Combinatorics ,010104 statistics & probability ,symbols.namesake ,0101 mathematics ,Anisotropy ,Mathematics ,Random field ,Mathematics::Commutative Algebra ,Packing dimension profile ,010102 general mathematics ,Gaussian random fields ,Range ,16. Peace & justice ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Packing dimension ,Metric space ,symbols ,60G15, 60G18, 28A80 ,Borel set ,[ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR] - Abstract
International audience; Let $X=\{X(t), t \in \R^N\}$ be a Gaussian random field with values in $\R^d$ defined by $$X(t) = \big(X_1(t), \ldots, X_d(t)\big), \qquad \forall \ t \in \R^N, $$ where $X_1, \ldots, X_d$ are independent copies of a centered real-valued Gaussian random field $X_0$. We consider the case when $X_0$ is anisotropic and study the packing dimension of the range $X(E)$, where $E\subseteq \R^N$ is a Borel set. For this purpose we extend the original notion of packing dimension profile due to Falconer and Howroyd (1997) to the anisotropic metric space $(\R^N, \rho)$, where $\rho(s, t) = \sum_{j=1}^N |s_j - t_j|^{H_j}$ and $(H_1, \ldots, H_N) \in (0, 1)^N$ is a given vector. The extended notion of packing dimension profile is of independent interest.
- Published
- 2011