Summary
Let X t be a real Gaussian process with stationary increments, mean 0, σ 2t =E[(X s+t−X s)2] If σ σ 2t behaves like t α as t ↺0, 0<α<1, the graph of a.e. sample function will have Hausdorff dimension 2 -α. This leads one to feel that the set of zeros of X t should have Hausdorff dimension 1 -α. This is shown to be true provided the process is stationary and satisfies additional assumptions.
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Orey, S. Gaussian sample functions and the Hausdorff dimension of level crossings. Z. Wahrscheinlichkeitstheorie verw Gebiete 15, 249–256 (1970). https://doi.org/10.1007/BF00534922
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DOI: https://doi.org/10.1007/BF00534922