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Hausdorff Measure and Uniform Dimension for Space-Time Anisotropic Gaussian Random Fields

Author

Listed:
  • Weijie Yuan

    (Zhejiang Gongshang University)

  • Zhenlong Chen

    (Zhejiang Gongshang University)

Abstract

Let $$X=\{ X(t), t\in \mathbb {R}^{N}\} $$ X = { X ( t ) , t ∈ R N } be a centered space-time anisotropic Gaussian random field in $$\mathbb {R}^d$$ R d with stationary increments, where the components $$X_{i}(i=1,\ldots ,d)$$ X i ( i = 1 , … , d ) are independent but distributed differently. Under certain conditions, we not only give the Hausdorff dimension of the graph sets of X in the asymmetric metric in the recurrent case, but also determine the exact Hausdorff measure functions of the graph sets of X in the transient and recurrent cases, respectively. Moreover, we establish a uniform Hausdorff dimension result for the image sets of X. Our results extend the corresponding results on fractional Brownian motion and space or time anisotropic Gaussian random fields.

Suggested Citation

  • Weijie Yuan & Zhenlong Chen, 2024. "Hausdorff Measure and Uniform Dimension for Space-Time Anisotropic Gaussian Random Fields," Journal of Theoretical Probability, Springer, vol. 37(3), pages 2304-2329, September.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:3:d:10.1007_s10959-024-01323-7
    DOI: 10.1007/s10959-024-01323-7
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    References listed on IDEAS

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    1. Wenqing Ni & Zhenlong Chen, 2021. "Hausdorff Measure of the Range of Space–Time Anisotropic Gaussian Random Fields," Journal of Theoretical Probability, Springer, vol. 34(1), pages 264-282, March.
    2. Chen, Zhenlong & Wang, Jun & Wu, Dongsheng, 2020. "On intersections of independent space–time anisotropic Gaussian fields," Statistics & Probability Letters, Elsevier, vol. 166(C).
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