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

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Abstract

Let \(X=\{ X(t), t\in \mathbb {R}^{N}\} \) be a centered space-time anisotropic Gaussian random field in \(\mathbb {R}^d\) with stationary increments, where the components \(X_{i}(i=1,\ldots ,d)\) 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.

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  1. School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou, 310018, China

    Weijie Yuan & Zhenlong Chen

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  1. Weijie Yuan
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  2. Zhenlong Chen
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Correspondence to Zhenlong Chen.

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This work was supported by the National Natural Science Foundation of China (12371150), Zhejiang Province Philosophy and Social Science Planning Routine Subject (24NDJC131YB) and the Management Project of "Digital+" Discipline Construction of Zhejiang Gongshang University (SZJ2022A012, SZJ2022B017) .

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Yuan, W., Chen, Z. Hausdorff Measure and Uniform Dimension for Space-Time Anisotropic Gaussian Random Fields. J Theor Probab (2024). https://doi.org/10.1007/s10959-024-01323-7

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