Dimension results for Gaussian vector fields and index-\(\alpha\) stable fields
type: Item
instance of: scholarly article
title: Dimension results for Gaussian vector fields and index-\(\alpha\) stable fields
zbMATH Open document ID: 0834.60040
author: Y. Xiao
published in: The Annals of Probability
publication date: 1995-09-17T00:00:00Z
review text: 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.
reviewed by: T. Inglot
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60G60
zbMATH DE Number: 796543
zbMATH Keywords: multiple points
zbMATH Keywords: Gaussian vector field
zbMATH Keywords: stationary increments
zbMATH Keywords: packing dimension
zbMATH Keywords: Hausdorff dimension
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176988387
OpenAlex ID: W2093912528
DOI: 10.1214/AOP/1176988387
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description: scientific article; zbMATH DE number 796543