## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 43, Number 2 (1972), 690-694.

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

#### Abstract

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.

#### Article information

**Source**

Ann. Math. Statist., Volume 43, Number 2 (1972), 690-694.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177692657

**Digital Object Identifier**

doi:10.1214/aoms/1177692657

**Mathematical Reviews number (MathSciNet)**

MR426167

**Zentralblatt MATH identifier**

0239.60035

**JSTOR**

links.jstor.org

#### Citation

Hendricks, W. J. Hausdorff Dimension in a Process with Stable Components--An Interesting Counterexample. Ann. Math. Statist. 43 (1972), no. 2, 690--694. doi:10.1214/aoms/1177692657. https://projecteuclid.org/euclid.aoms/1177692657