Dimension results for Gaussian vector fields and index-\(\alpha\) stable fields
Let \(X(t)\) be \((N,d)\) Gaussian vector field of index \(\alpha = (\alpha_1, \ldots, \alpha_d)\), i.e. such that each of \(d\) components of \(X\) has stationary increments and \(E(X_j (t) - X_j (0))^2\) ``behaves like'' \(|t |^{2 \alpha_j}\) as \(|t |\to 0\), \(0 < \alpha_j \leq 1\), \(j = 1, \ldots, d\). Hausdorff dimension of the image \(X(E)\) and graph \(\text{Gr} X (E)\) for any compact set \(E\) is found. Moreover, packing dimension is obtained for \(E = [0,1]^N\). Analogous statements are proved for index-\(\alpha\) stable fields. These results generalize some known theorems and correct some errors in the literature. Finally, let \(X(t)\) be an index-\(\alpha\) stable field. \(x \in R^d\) is called a \(k\)-multiple point of \(X\) if \(X(t_1) = \cdots = X(t_k) = x\) for some distinct \(t_1, \ldots, t_k \in R^N\). It is shown that for \(Nk > (k - 1) (\alpha_1 + \cdots + \alpha_d)\) the field \(X(t)\) has \(k\)-multiple points with positive probability and in this case the Hausdorff dimension of the set of \(k\)-multiple times is given. This generalizes results of Cuzick for Gaussian fields.
scientific article; zbMATH DE number 796543