Open Access
February, 1977 Hausdorff Dimension and Gaussian Fields
Robert J. Adler
Ann. Probab. 5(1): 145-151 (February, 1977). DOI: 10.1214/aop/1176995900


Let $X(t)$ be a Gaussian process taking values in $R^d$ and with its parameter in $R^N$. Then if $X_j$ has stationary increments and the function $\sigma^2(t) = E\{|X_j(s + t) - X_j(s)|^2\}$ behaves like $|t|^{2\alpha}$ as $|t| \downarrow 0, 0 < \alpha < 1$, the graph of $X$ has Hausdorff dimension $\min \{N/\alpha, N + d(1 - \alpha)\}$ with probability one. If $X$ is also ergodic and stationary, and if $N - d\alpha \geqq 0$, then the dimension of the level sets of $X$ is a.s. $N - d\alpha$.


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Robert J. Adler. "Hausdorff Dimension and Gaussian Fields." Ann. Probab. 5 (1) 145 - 151, February, 1977.


Published: February, 1977
First available in Project Euclid: 19 April 2007

zbMATH: 0366.60050
MathSciNet: MR426123
Digital Object Identifier: 10.1214/aop/1176995900

Primary: 60G15
Secondary: 60G17

Keywords: capacity , Gaussian fields , Hausdorff dimension , Level sets

Rights: Copyright © 1977 Institute of Mathematical Statistics

Vol.5 • No. 1 • February, 1977
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