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February 2022 The Hausdorff measure of the range and level sets of Gaussian random fields with sectorial local nondeterminism
Cheuk Yin Lee
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Bernoulli 28(1): 277-306 (February 2022). DOI: 10.3150/21-BEJ1342

Abstract

We determine the exact Hausdorff measure functions for the range and level sets of a class of Gaussian random fields satisfying sectorial local nondeterminism and other assumptions. We also establish a Chung-type law of the iterated logarithm. The results can be applied to the Brownian sheet, fractional Brownian sheets whose Hurst indices are the same in all directions, and systems of linear stochastic wave equations in one spatial dimension driven by space–time white noise or colored noise.

Acknowledgements

The author wishes to thank Thomas Mountford and Yimin Xiao for stimulating discussions and helpful comments that have led to improvements in the paper.

Citation

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Cheuk Yin Lee. "The Hausdorff measure of the range and level sets of Gaussian random fields with sectorial local nondeterminism." Bernoulli 28 (1) 277 - 306, February 2022. https://doi.org/10.3150/21-BEJ1342

Information

Received: 1 December 2020; Revised: 1 March 2021; Published: February 2022
First available in Project Euclid: 10 November 2021

Digital Object Identifier: 10.3150/21-BEJ1342

Keywords: Brownian sheet , Gaussian random fields , harmonizable representation , Hausdorff measure , Local nondeterminism

Rights: Copyright © 2022 ISI/BS

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Vol.28 • No. 1 • February 2022
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