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Hausdorff Dimension in a Process with Stable Components--An Interesting Counterexample

# Hausdorff Dimension in a Process with Stable Components--An Interesting Counterexample,10.1214/aoms/1177692657,The Annals of Mathematical Statistics,W

Hausdorff Dimension in a Process with Stable Components--An Interesting Counterexample
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.
Journal: The Annals of Mathematical Statistics , vol. 43, no. 1972, pp. 690-694, 1972
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## Citation Context (3)

• ...These were introduced in the one-parameter case by Pruitt [PT69], and investigated further by Hendricks [H70, H72, H73, H74]...

### Davar Khoshnevisan, et al. Level sets of the stochastic wave equation driven by a symmetric Lévy ...

• ...For the special case of a Le´ vy process X with stable components in R d ; the Hausdorff dimension of the range X ð½0; 1�Þ was studied by Pruitt and Taylor [22] and then extended by Hendricks [8,9] who determined the Hausdorff dimension of X ðEÞ; where E � Rþ is a fixed Borel set...
• ...The upper bounds in Theorems 3.1 and 3.2 are proved by using Lemmas 3.3 and 3.4 and a covering argument which goes back to Pruitt and Taylor [22] and Hendricks [8,9]; while the lower bounds are proved by using Lemma 3.7 and (2.12)...

### Mark M. Meerschaert, et al. Dimension results for sample paths of operator stable Lévy processes

• ...We note that not all independent increment processes in IR d(d > 2) have an index e which makes (0.1) valid: the first counterexample is due to Hendricks [7]...

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