Hausdorff Dimension and Gaussian Fields

Adler, Robert J.

Ann. Probab., Tome 5 (1977) no. 6, p. 145-151 / Harvested from Project Euclid

Résumé

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. $N - d\alpha$.

Détail

Publié le : 1977-02-14
Classification:  Gaussian fields,  level sets,  Hausdorff dimension,  capacity,  60G15,  60G17