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April, 1972 Hausdorff Dimension in a Process with Stable Components--An Interesting Counterexample
W. J. Hendricks
Ann. Math. Statist. 43(2): 690-694 (April, 1972). DOI: 10.1214/aoms/1177692657


Let $X_{\alpha_1}(t)$ and $X_{\alpha_2}(t)$ be independent stable processes in $R_1$ of stable index $\alpha_1$ and $\alpha_2$ respectively, where $1 < \alpha_2 < \alpha_1 \leqq 2$. Let $X(t) \equiv (X_{\alpha_1} (t), X_{\alpha_2}(t))$ be a process in $R_2$ formed by allowing $X_{\alpha_1}$ to run on the horizontal axis and $X_{\alpha_2}$ on the vertical axis; $X(t)$ is called a process with stable components. The Blumenthal-Getoor indices of $X(t)$ satisfy $\alpha_2 = \beta" < \beta' = 1 + \alpha_2 - \alpha_2/\alpha_1 < \beta = \alpha_1$. Denote by $\dim E$ the Hausdorff dimension of $E$. It is shown that if $E = \lbrack 0, 1\rbrack$ and $F$ is any fixed Borel set for which $\dim F \leqq 1/\alpha_1$ then (with probability 1) we have $\dim X(E) = \beta' \dim E$ and $\dim X(F) = \beta \dim X(F)$. This shows that the results of Blumenthal and Getoor (1961) for the bounds on $\dim X(E)$ for arbitrary processes $X$ and fixed Borel sets $E$ are the best possible, and that their conjecture that $\dim X(E) = \dim X\lbrack 0, 1\rbrack \cdot \dim E$ is incorrect.


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W. J. Hendricks. "Hausdorff Dimension in a Process with Stable Components--An Interesting Counterexample." Ann. Math. Statist. 43 (2) 690 - 694, April, 1972.


Published: April, 1972
First available in Project Euclid: 27 April 2007

zbMATH: 0239.60035
MathSciNet: MR426167
Digital Object Identifier: 10.1214/aoms/1177692657

Rights: Copyright © 1972 Institute of Mathematical Statistics


Vol.43 • No. 2 • April, 1972
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