Hausdorff Measure of the Range of Space–Time Anisotropic Gaussian Random Fields

  • Wenqing Ni
  • Zhenlong ChenEmail author


Let \(X=\{X(t)\in {{\mathbb {R}}}^d, t\in {{\mathbb {R}}}^N\}\) be a centered space–time anisotropic Gaussian random field with stationary increments, whose components are independent but may not be identically distributed. Under certain mild conditions, we determine the exact Hausdorff measure function for the range \(X([0,1]^N)\). Our result extends those in Talagrand (Ann Probab 23:767–775, 1995) for fractional Brownian motion and Luan and Xiao (J Fourier Anal Appl 18:118–145, 2012) for time-anisotropic and space-isotropic Gaussian random fields.


Space–time anisotropic Gaussian random fields Strong local nondeterminism Range Hausdorff measure 

Mathematics Subject Classification (2010)

60G15 60G17 60G60 



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Authors and Affiliations

  1. 1.School of Statistics and MathematicsZhejiang Gongshang UniversityHangzhouChina
  2. 2.School of ScienceJimei UniversityXiamenChina

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